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This text book was oimed by Janes E. j
Gillespie, \itio was the first Division Engin-j
eer of the Milwaukee Division of the State
Highway Oonmissionf establishing that office
in Nay of 1916. Donated in his memory by
his nephew, James E. Meier, Waukesha Dis-
trict Engineer from 1957 to 1965.
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'^-^'^''UJty' ^yJ<..&y^U^
KURT F. WENDT 1 "?'''^.RY
COLLEGE OH F^'C'.:4ciRING
UNIVEr.3ITY OF WISCONSJN
MADISON, Wl 53706
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WORKS OF PROF. J. B. JOHNSON
PUBLISHED BY
JOHN WILEY & SONS
Theory and Practice In the Designing of Modem
Pramed Structures.
Profusely illustrated with figures in the text and
full-paee plate, including many half-tones. 410^
cloth, |lO.CK>.
The Theory and Practice of Surveying.
Designed for the use of Surveyors and Engineers
generally, but especially for the use of Students
in Engineering. 8vo, cloth, $4.00.
The Materials of Coostmctlofi.
A treatise for Engineers on the Strength and other
Properties of Engineering Materials. All the Data
newly compiled trom the latest home and foreign
tests, l^rge 8vo, 800 pages, 6so illustrations, it
plates, complete index $6.00.
Pubiuked by Engtnetring Ntwt:
Engineering Contracts and Speciflcatlons.
Including a brief synopsis of the Law of Contracts,
and illustrative examples of the General and Technical
Clauses of various kinds of Engineering Specifications.
Designed for the use of Students, Engineers and
Contractors. - Engineering News Publishmg Co., St.
Paul Building, New York. Price $3.00, postpaid.
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THE THEORY AND PRACTICE
OF
SURVEYING.
DESIGNED FOR THE USE OF
SURVEYORS AND ENGINEERS GENERALLY.
BUT ESPECIALLY FOR THE USE OF
Students in Engineering
J. B. JOHNSON, C.E.,
Lafe Dean of the College of Meclianics and Engineering of the Univ» of Wisconsin;
Formerly Civil Engineer on the U» S. Lake and Mississippi River
Surveys ; Member Inst. Civil Engineers; Member of the
American Society of Civil Engineers ^ etc., etc,
SIXTBBJVTH EPITION, REVISED AND ENLARGED.
SEVENTH THOUSAxVD.
NEW YORK :
JOHN WILEY & SONS.
Ix>ni>on: CHAPMAN & HALL, Limited.
1904.
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Copyright, ib86, 1900
By T. B. JOHNSON.
PRESS OP
BR AUN WORTH & CO.
BOOKBINDERS AND PRINTERS
BROOKLYN, N. V.
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J^^l ^l}C
PREFACE TO THE SIXTEENTH EDITION.
The principal changes in this edition are the following :
(i) Many changes in Chap. XIV on Geodetic Surveying,
especially concerning base-line measurements and precise level-
ing, to adapt these portions to the recent greatly modified
practices of the U. S. C. & G. Survey.
(2) A new Table XII has been computed (pp. 814, 815),
for the years 1902-1910. This table gives the azimuth of
Polaris for all hour-angles, and for all latitudes from 30® to 50®.
(3) Several minor changes and corrections have been made,
notably on p. 468, in reference to the shrinkage of earthwork.
J. B. J.
Madisou, Jannary, 1902.
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PREFACE TO THE FIFTEENTH EDITION.
The principal changes in this edition are the following :
1. A new field method of determining the refraction correc-
tion to apply to the declination setting in solar azimuth work
is given in Art $^a. This method was devised by G. C. Com-
stock, Professor of Astronomy in the University of Wisconsin.
2. A description of the slide-rule, with illustrative examples
of its use, is given in Art. 156^:. While the author has long
been a constant user of slide-rules of all kinds, he had not
thought to include it in a description of surveying instruments.
It is now introduced here simply because its use is not taught
elsewhere in our engineering schools.
3. Various improvements in the field methods of surveying
with the transit and stadia, in order to increase the accuracy of
this kind of work. These are found in the new articles 200a,
201, 218, 2i8a, all of which have been prepared by L. S. Smith,
Assistant Professor of Topographical and Geodetic Engineer-
ing in the University of Wisconsin.
4. Chapter XI, on Mining Surveying, has been entirely re-
written by Prof. Robert S. Stockton, E.M., of the Colorado
State School of Mines, Golden, Col., and by Mr. Edward P.
Arthur, Jr., E.M., U. S. Deputy Mineral Surveyor, Cripple
Creek, Col. Both these gentlemen are accomplished mining
and mineral land surveyors, and one of them is an experienced
teacher. It is believed, therefore, that this chapter has been
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PREFACE TO THE FIFTEENTH EDITION. V
much improved and is better suited, both for instruction and
for field purposes, than the former chapter on this subject.
5. A new Appendix B, being the latest Manual of Instruc-
tions for the Survey of Mineral Lands. This has been brought
up to date, 1899, ^^^ ^^ much more full and complete than the
former Appendix B.
6. A new Appendix I, which is a reprint of the latest Rules
for Restoring Lost Corners as issued by the General Land
Office at Washington. This is a very important addition.
It is thought that these changes and additions will consider-
ably enhance the value of a work which has secured and still
retains the favor of both the surveyors and teachers of survey-
ing at home and abroad to a degree far beyond the author's
fondest hopes when he undertook the work some fifteen years
ago.
The author's desire and purpose to keep this work fully
abreast of the best American practice is his only excuse for the
numerous additions and changes which have been made in the
various successive editions.
J.B.J.
Madison, Wis., January, 1900.
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NOTE.
In the second thousand of the fifteenth edition a new
method of finding azimuth from an observation on Polaris at
any hour is given in Art. $Sia on page 569. This method and
the tables used in it were devised by G. C. Comstock, Professor
of Astronomy in the University of Wisconsin. For the obtain-
ing of an azimuth within an error of one minute the author of
this work recommends this method as superior to all others
when it is inconvenient to make the observation on Polaris when
near elongation. The method is, however, as suitable for an
observation at elongation as at any other time. The method
heretofore used in this work for a similar purpose, which was
taken from the Manual of Instructions used by the Commis-
sioner of the General Land Office, is also retained and now
forms Art. 381*.
A three-place table of logarithms of numbers and of trigo-
nometrical functions, all on one double-page inset, has been
added as Table la, p. 756. A corresponding pocket edition of
this table also will be inserted in all copies of the sixteenth
and subsequent editions of this work. These three-place tables
will be found sufficiently accurate for many of the computations
required in surveying. They are of especial value in laboratory
computations.
Certain other minor changes and corrections have been
made, including the computation of new tables for the elonga-
tion and culmination of Polaris and 51 Cephei, beginning with
the year 1901. These changes have been made on pages 32,
560, and 561.
VI
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PREFACE TO THE TWELFTH EDITION.
Since the issue of the seventh edition of this work, in
1890, there was added in the tenth edition (1892) Appendix
F, being the instructions for field work issued by the Missis-
sippi River Commission, and there was added in the eleventh
edition Appendix G, upon the Essential Requirements of Sur-
veys and Maps, and upon the Ownership of Surveys, by Prof.
Wm. G. Raymond. There is now added to the twelfth edi-
tion Appendix H, containing the Michigan Instructions for
the Making and Filing of Town, City, and Village Plats, with
various accompanying legal forms. While the laws of other
States will not require the particular procedure in these mat-
ters, here laid down, the general following of these instructions
will greatly improve the current practice everywhere. All
students and young surveyors are urged to read these two last
appendices with care.
There has also been added a Table of Azimuths of Polaris,
from 189S to 1910; and a method of finding the meridian by a
single measurement of the altitude of the sun (p. 103^), this
being the most convenient method to use, but one which had
strangely fallen into disuse. Also a new isogonic chart show-
ing magnetic declinations in the United States for 1900, and
on pp. 2Sa to 28^/, a list of annual changes in the declination
for the years 1895 and 1900, both of these being furnished
through the courtesy of the Superintendent of the United
States Coast and Geodetic Survey, in advance of their publica-
tion by that office. The Swiss method of barometric levelling
between points of known elevation is also added to this edition
(p. 137), and a few other minor changes and corrections are
made. J. B. J.
September, 1896.
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PREFACE TO THE SEVENTH EDITION.
In each new edition of this volume some additions, cor-
rections, and minor changes have been made. In the present
edition there are so many changes and additions that they
deserve to be specially mentioned.
To Part I., on Surveying Instruments, have been added
descriptions and cuts of the architect's level, new level-rod tar-
gets and bubbles. Wood's double sextant, and the cross-section
polar protractor used in the New York aqueduct tunnel.
The table of Magnetic Declination Formulae, on pages 25 to
28 inclusive, has been replaced entire with the new table issued
by the United States Coast and Geodetic Survey, 1890, and
Plate I. has been redrawn and brought down to 1890.
The chapter on Land Surveying has been entirely recast.
A considerable amount of new matter on the subject of monu-
ments, and the principles and laws governing the re-survey of
lands, have been added. The author wishes here to acknowledge
his debt to Bellows and Hodgeman's little work on Land Sur-
veying for much of the original matter from which he has
deduced his general rules. In that work decisions are ab-
stracted, and references given to the cases themselves, and the
land surveyor would do well to obtain a copy of this valuable
work. In preparing these general rules the author has had in
mind the student and young surveyor rather than the ex-
perienced practitioner; and the reader must remember that
judicial decisions are judgments on particular cases, and gener-
-ilizations from such decisions must be made and received with
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PREFACE. ix
caution. The rules here laid down apply rather to the inexact
methods of the compass and chain than to those of the transit
and steel tape.
The description of the United States Land Surveys has
been entirely re-written and expanded, and an appendix added
giving the location of all the principal meridians and accom^
panying base lines which have been used in laying out the
public lands.
A method of running out parallels of latitude, with suitable
tables, has been added, and also tables and descriptions by
which an observation for azimuth may be made on Polaris at
any hour. This latter table has but recently been computed,
and is published in the last edition (1890) of the Manual of
Instructions issued by the Commissioner of the General Land
Office, Washington, D. C. By means of this table the great
objections to stellar observations for azimuth are removed, as
they may be made at any hour, and all tedious computations
are avoided.
A description of Porro's Telescope has also been added to
the chapter on Topographical Surveying. This telescope
reads distances by stadia correctly from the center of the
instrument instead of from a point in front of the objective.
It is not now manufactured in this country, but it may again
come into use,
J. B. J.
Decembbr, 189O.
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PREFACE TO THE FIRST EDITION.
No apology is necessary for the appearance of a new book
on Surveying. The needs of surveyors have long been far be-
yond the accessible literature on this subject, to say nothing of
that which has heretofore been formulated in text-books The
author's object has been to supply this want so far as he was
able to do it.
The subject of surveying, both in the books and in the
schools, has been too largely confined to Land Surveying, The
engineering graduates of our technical schools are probably
called upon to do more in any one of the departments of
Railroad, City, Topographical, Hydrographical, Mining, or
Geodetic Surveying than in that of Land Surveying. Some
of these subjects, as for example City, Geodetic, and Hy-
drographical Surveying, have not been formulated hitherto,
in any adequate sense, in either English or any other
language, to the author's knowledge. In the case of Geodetic
Surveying there has been a wide hiatus between the matter
given in text-books and the treatment of the subject in works
on Geodesy and in special reports of geodetic operations. The
latter was too technical, prolix, and difficult to give to stu-
dents, while the former was entirely inadequate to any rea-
sonable preparation for this kind of work on even a small
scale. The subjects of City and Hydrographical Surveying as
here presented are absolutely new.
Part L treats of the adjustment, use, and care of all kinds
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PREFACE. X'
of instruments used in surveying, either in field or office.* In
describing the adjustments of instruments the object has been
to present to the mind of the reader the geometrical relations
from which a rule or method of adjustment would naturally
follow. The author has no sympathy with descriptions of ad-
justments as mechanical processes simply to be committed to
memory, any more than he has with that method of teaching
geometry wherein the student is allowed to memorize the
demonstration.
Many surveying instruments not usually described in books
on surveying are fully treated, such as planimeters, panto-
graphs, barometers, protractors, etc. The several sets of prob.
lems given to be worked out by the aid of the corresponding
instruments are designed to teach the capacity and limitations
of such instruments, as well as the more important sources of
error in their use. This work is such as can be performed
about the college campus, or in the near vicinity, and is sup-
posed to be assigned for afternoon or Saturday practice while
the subject is under consideration by the class. More ex-
tended surveys require a special field-season for their success-
ful prosecution.f
The methods of the differential and integral calculus have
been sparingly used, as in the derivation of the barometric for-
mula for elevations, and of the LM Z formulae in Appendix
D. Such demonstrations may have to be postponed to a later
period of the course.
i
* Certain special appliances, as for example heliotropes* filar micrometenL
current meters, etc, are treated in the subsequent chapters.
f At Washington University all the engineering Sophomores go into the
field for four we^ks at the end of the college year, and make a general land
and topographical sorvey, &uch as shown in Plate II. At the end of the Juniot
year the dvil-engineering students go again for fojr weeks, making then a
geodetic and railroad survey. Some distant region is selected where the
ground, boarding facilities, etc., are suitable.
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XI 1 PREFACE.
Part 11. includes descriptions of the theory and practice ol
Surveying Methods in the several departments of Land, Topo-
graphical, Railroad, Hydrographical, Mining, City, and Geo-
detic Surveying ; Surveys for the Measurement of Volumes ;
and the Projection of Maps, Map Lettering, and Topographic
cal Signs. The author has tried to treat these subjects in a
concise, scientific, and practical way, giving only the latest and
most approved methods, and omitting all problems whose
only claim for attention is that of geometrical interest.
In treating the trite subject of Land Surveying many prob-
lems which are more curious than useful have been omitted,
and several new features introduced. The subjects of com-
puting areas from the rectangular co-ordinates, and the supply-
ing of missing data, are made problems in analytical geometry,
as they should be. A logarithmic Traverse Table for every
minute of arc from zero to 90°, arranged for all azimuths from
zero to 360**, to be used in connection with a four-place loga-
rithmic table, serves to compute the co-ordinates of lines when
the transit is the instrument used. A traverse table com-
puted for every 15 minutes of arc is no longer of much value.
The isogonic declination curves shown on Plate L will be found
to embody all the accessible data up to 1885, and are reduced
from the U. S. Coast Survey chart. Appendix A will be
found of great value as outlining the Judicial Functions of the
Surveyor by the best possible authority.
The chapter on Mining Surveying was written by Mr. C
A.. Russell, C.E., U. S. Deputy Mineral Surveyor of Boulder,
Colorado.* He has had an extended experience in Hydro-
graphic Surveying, in addition to many years' practice in sur-
veying mines and mining claims.
* This chapter has now been replaced by another by Prof. R. S. Stocktoiv
of the Colorado School of Mines.
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PREFACE. Xlll
The chapter on City Surveying was written by Mr. Wm,
Bouton, C.E., City Surveyor of St. Louis, Mo. Mr. Bouton
has done a large proportion of the city surveying in St. Louis
for the last twenty years, and has gained an enviable reputa*
tion as a reliable, scientific, and expert surveyor.
It is believed that the ripe experience of these gentlemen
which has been embodied in their respective chapters will ma*
terially enhance the value of the book.
The author also desires to acknowledge his indebtedness to
his friend H. S. Pritchett, Professor of Astronomy in Wash-
ington University,* for valuable assistance in the preparation
ot the matter on Time in Chapter XIV.
Although the theorems and the notation of the method of
least squares are not used in this work, yet two problems are
solved by what is called the method of the arithmetic mean
(which, when properly defined, is the same as the method of
least squares), which will serve as a good introduction to the
study of the method of least squares. These problems are the
Rating of a Current-meter, in Chapter X., and the Adjustment
of a Quadrilateral, in Chapter XIV. The author has found
that such solutions as these serve to make clear to the mind
of the student exactly what is accomplished by the least-
square methods of adjusting observations.
The chapter on Measurement of Volumes is not intended
to be an exhaustive treatment of the subject of earthwork, but
certain fundamental theorems and relations are established
which will enable the student to treat rationally all ordinary
problems. The particular relation between the Henck pris*
moid and the warped-surface prismoid is an important one,
but one which the author had nowhere found.f
* Since made Superintendent of the U. S. Coast and Geodetic Survey and
now (1900) President of the- Massachusetts Institute of Technology.
f The author's attention has since been called to the fact that this relation
was published in Henck's " Field Book'' in 1881.
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XIV PREFACE,
An earthwork table (Table XI.) has also been prepared
which gives volumes directly, without correction, for the
warped-surface prismoid. The author has no knowledge that
such a table has ever been prepared before.
A former work by the author on Topographical Surveying
by the Transit and Stadia is substantially included in this
book.
The methods recommended for measuring base-lines with
steel-tapes are new; but they have been thoroughly tested,
and are likely to work a material change in geodetic methods.
The author wishes to acknowledge his obligations to many
instrument-manufacturers for the privilege they have very
kindly accorded to him of having electrotype copies made from
the original plates, for many of the cuts of instruments given
throughout the book ; persons familiar with the valuable cata-
logues published by these firms will recognize the makers
among the following: W. & L. E. Gurley, Troy, N. Y. ; Buff
& Berger, Boston, Mass. ; Fauth & Co., Washington, D. C. ;
Queen & Co. and Young & Sons, Philadelphia, Pa.; Keuffel
& Esser, New York; and F. E. Brandis Sons & Co., Brook-
lyn, N. Y. Also to Mr. W. H. Searles for the privilege of
using copies of plates from his Field-book for Tables I., VI.,
and VII.
Hoping this work will assist in lifting the business of sur-
veying to a higher professional plane, as well as to enlarge its
boundaries, the author submits it to surveyors and engineers
generally, but especially to the instructors and students in our
engineering schools, for such crucial tests as the field and the
class-room only can give.
J.B.J.
St. Louis, Sept. 33, x8S6.
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TABLE OF CONTENTS.
PAGl
iNTRODUCriON... •••...».•. I
BOOK L
SURVEYING INSTRUMENTS.
CHAPTER I.
INSTRUMENTS FOR MEASURING DISTANCES.
Thb Chain :
I. The Engineer's Chain 5
%. Gunter's Chain 5
3. Testing the Chain 6
4. The Use of the Chain 8
The Steel Tape :
5. Varieties 9
6. The Use of Steel Tapes 10
Exercises with the Chain:
7-17. Practical Problems 11, la
CHAPTER II.
INSTRUMENTS FOR DETERMINING DIRECTIONS.
The Compass :
18. The Surveyor's Compass described 13
19. The General Principle of Reversion 15
2a To make the Plate perpendicular to the Axis of the Socket 16
31. To make the Plane of the Bubbles perpendicular to the Axis of the
Socket 16
22. To adjust the Pivot to the Centre of the Graduated Circle 16
XV
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XVl CONTENTS,
PAGI
«3. To straighten the Needle 17
24. To make the Plane of the Sights normal to the Plane of the Bubbles. 17
25. To make the Diameter through the Zero-graduations lie in the Plane
of the Sights 17
26. To remagnetize the Needle 18
27. The Construction and Use of Verniers < • 18
The Declination of the Needle:
28. The Declination defined 20
29. The Daily Variation 20
30. The Secular Variation 21
31. Isogonic Lines 23
32. Other Variations of the Declination 29
33. To fi' d the Declination of the Needle 29
Use of th2 Needle Compass :
34. The Use of the Compass 34
35. To set off the Declination 36
36. Local Attractions 36
37. To establish a Line of a Given Bearing 37
38. To find the True Bearing of a Line. 37
39. To retrace an Old Line 37
The Prismatic Compass :
40. The Prismatic Compass described • 38
Exercises :
41-44. Exercises for the Needle Compass •••..« 38, 39
The Solar Compass :
45. The Burt Solar Compass •• 39
46. Adjustment of the Bubbles 41
47. Adjustment of the Lines of Collimation 41
48. Adjustment of the Declination Vernier 42
49. Adjustment of the Vernier on the Latitude Arc 43
50. Adjustment of Terrestrial Line of Sight to the Plane of the Polar
Axis 43
Use of the Solar C jIPass :
51. Conditions requiring its Use 44
52. To find the Declination of the Sun 44
53. To correct the Declination for Refraction • ••.. .••••••• 45
53rt. A Field Determination of the Refraction Correction 48^
54. Errors in Azimuth due to Errors in the Decl. and Lat Angles 49
55. Solar Attachments 52
Exercises with the Solar Compass :
56-59. Practical Problems 53, 54
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CONTENTS, xvii
CHAPTER 411.
INSTRUMENTS FOR DETERMINING HORIZONTAL LINES.
PAGE
Plumb-line and Bubble:
60. Their Universal Use in Surveying and Astronomical Work 55
61. The Accurate Measurement of small Vertical Angles 58
62. The Angular Value of one Division of the Bubble 58
63. General Considerations 59
1 HE Engineer's Level :
64. The Level described 60
65. Adjustment of Line of Sight and Bubble Axis to Parallel Positions. 63
66. Lateral Adjustment of Bubble 67
67. The Wye Adjustment 67
68. Relative Importance of Adjustments t . . . . 68
69. Focussing and Parallax 68
tx^. The Architect's Compass Level ,•••• 69^
70. The Levelling-rod 69^
71. The Use of the Level 71
DiFFEEENTIAL LEVELLING:
72. Differential Levelling defined 72
73. Length of Sights 73
74. Bench-marks 75
75. The Record 76
76. The Field work 76
Profile Levelling:
77. Profile Levelling defined ,... . 77
78. The Record 78
Levelling for Fixing a Grade :
79. The Work described 81
The Hand Level:
80. Locke's Hand Level 81
Exercises with the Level :
81^5. Practical Problems 82
CHAPTER IV. '
INSTRUMENTS FOR MEASURING ANGLES.
. The Transit : the transit.
86. The Engineer's Transit described 83
87. The Adjustments Slated . , 86
88. Adjustment of Plate Bubbles 86
89. Adjustment of Line of Collimation 87
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xviu CONTENDS.
PACE
90. Adjustment of the Horizooul Axis 87
91. Adjustment of the Telescope Bubble 89
92. Adjustment of Vernier on Vertical Circle 89
93. Relative Importance of Adjustments 89
Instrumental Conditions affecting Accurate Measurements :
94. Eccentricity of Centres and Verniers 90
95. Inclination of Vertical Axis 91
96. Inclination of Horizontal Axis 92
97. Error in CoUimation Adjustment 93
The Use of the Transit :
98. To measure a Horizontal Angle 93
99. To measure a Vertical Angle 94
100. To run out a Straight Line ... 95
loi. Traversing 97
The Solar Attachment :
102. Various Forms described 99
103. Adjustments of the Saegmuller Attachment 102
The Gradienter Attachment :
104. The Gradienter described..... 104
The Care of the Transit :
105. The Care of the Transit 104
Exercises with the Transit •
106-114. Practical Problems 105-107
The Sextant :
115. The Sextant described 108
116. The Theory of the Sextant no
117. The Adjustment of the Index Glass in
118. The Adjustment of the Horizon Glass in
119. The Adjustment of the Telescope to the Plane of the Sextant in
120. The Use of the Sextant 112
Exercises with the Sextant :
121. 121/1. Practical Problems 112,113
122. Wood *s Doable Sextant ,« 113
122a. The Cross-section Polar Protractor • 114
CHAPTER V.
THE PLANE TABLE.
123. The Plane Table described •••••••• 117
124. Adjustment of the Plate Rubbles 119
125. Adjustment of Horizontal Axis • 1x9
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CONTENTS. xix
PAGB
126. Adjustment of Vernier and Bubble to Telescopic Line of Sight. . . 119
The Use of the Plane Table:
127. General Description of its Use 12a
128. Location by Resection 123
129. Resection on Three Known Points 123
130. Resection on Two Known Points 124
131. The Measurement of the Distances by Stadia 125
Exercises with the Plane Table:
132-135. Practical Problems 136
CHAPTER VI.
ADDITIONAL INSTRUMENTS USED IN SURVEYING AND PLOTTING.
The Aneroid Barometer;
136. The Aneroid described 127
137. Derivation and use of Barometric Formulae 128
138. Use of the Aneroid 136
The Pedometer :
139. The Pedometer described 137
The Length of Men*s Steps 138
The Odometer:
140. Description and Use 139
The Clinometer :
141. Description and Use 141
The Optical Square :
142. Description and Use 142
The Planimeter:
143. Description 143
144. Theory of the Polar Planimeter 144
145. To find the Length of Arm to use 150
146. The Suspended Planimeter 152
147. The Rolling Planimeter 152
148. Theory of the Rolling Planimeter 154
149. To Test the Accuracy of a Planimeter 157
15a The Use of the Planimeter 158
151. Acoiracy of Planimeter Measurements 160
The Pantograph :
152. Description and Theory 161
Various Styles of Pantographs 163
153. Use of the Pantograph 165
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PigOTRACTORS .
154. Various Styles described 166
Parallel Rulers:
155. Description and Use 169
Scales :
156. Various Kinds described 169
156.7. The Pocket Slide Rule 17X
BOOK II.
SUR VE YING ME THODS.
CHAPTER VII.
LAND-SURVEYING.
157. Land Surveying defined ••••••••••••••••••••••• 17)
158. Laying out Land • 172
159. Land Monuments 173
160. Significance and Authority of Monuments 174
161. Lost Monuments 175
The United States System of Laying out the Pubuc Lands :
162. The Extent of the System 176
163. The Reference Lines • 177
164. The Division into Townships 178
165. The Division into Sections 178
> 166. The Convergence of the Meridians 179
167. Corner Monuments 181
168. The Subdivision of Sections 183
169. The Running of Parallels 185
Finding the Area or Superficial Contents of Land when the
Limiting Boundaries are given :
170. The Area defined 187
By Triangular Subdivision :
171. By the Use of the Chain alone 188
172. By the Use of the Compass or Transit and Chain 189
173. By the Use of the Transit and Stadia 189
From Bearing and Length of the Boundary Lines :
174. The Common Method 189
175. The Field Notes 190
Z76. Method cf Computation stated • 193
177. Latitudes, Departures, and Meridian Distances ••• -.'•••• tof
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CONTENTS. XXI
PAGE
178. Computing the Latitudes and Departures . . . . ; 195
179. Balancing the Survey 198
180. The Error of Closure 201
181. The Form of Reduction 202
183. Correction from Erroneous Length of Chain 205
Area from the Rectangular Co-ordinates of the Corners:
183. Conditions of Application of the Method 208
184. Theory of the Method 209
1S5. The Form of Reduction 21:
Supplying Missing or Erroneous Data :
186. Equations for Supplying Missing Data — Four Cases. 211
Plotting:
187. Plotting the Survey 2x6
Irregular Areas :
188. The Method by Offsets at Irregular Intervals 216
189. The Method by Offsets at Regular Intervals 218
T"«ie Sl'bdivision of Land :
190. The Problems of Infinite Variety 221
191. To cut from a Given Tract of Land a Given Area by a Right Line
starting from a Given Point in the Boundary 221
192. To cut from a Given Tract of Land a Given Area by a Right Line
running in a Given Direction 223
PRINaPLES AND LAWS BEARING ON THE Re-SURVEY OF PRIVATE LaNDS :
193. The Problem Stated 228
194. The Interpretation of Descriptions in Deeds and the Identification
of Boundaries 229
195. Water Boundaries and Meandered Lines 232
196. Treatment of Surplus and Deficiency 233
Examples in Land Surveying 234
CHAPTER VIII.
TOPOGRAPHICAL SURVEYING BY THE TRANSIT AND STADIA.
197. Topographical Survey defined 237
198. Available Methods 237
199. Method by Transit and Stadia stated 238
Theory of Stadia Measurements :
20a Fundamental Relations 238
200tj. The Use of an Interval Factor 244
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XXII CONTENTS.
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201. A Simple Determination of the Wire-interval 245a
202. The Prevention of Systematic Errors of Stadia Measurements 245 r
203. Adaptation of Formulae to Inclined Sights 246
204. Description and Use of the Stadia Tables 248
205. Porro's Telescope, having the Vertex of the Reading Angle at the
Center of the Instrument • • 249
The Instruments :
206. The Transit 251
207. Setting the Cross-wires 252
208. Graduating the Stadia Rod 253
General Topographical Surveying :
209. Topography 257
210. Methods of Field Work 257
211. Reduction of the Notes ' 265
212. Plotting the Stadia Line. ... 268
213. Check Readings 269
214. Plotting the Side Readings 270
215. Contour Lines 275
216. ThcFinalMap 278
217. Topographical Symbols 279
218. Accuracy of the Stadia Method 379
CHAPTER IX.
RAILROAD TOPOGRAPHICAL SURVEYING.
219. Objects of the Survey 381
220. The Field Work 28 1
221. The Maps 283
222. Plotting the Survey 285
223. Making the Location on the Map 287
234. Another Method « 391
CHAPTER X.
HYDROGRAPHIC SURVEYING.
335. Hydrographic Surveying defined 393
The Location of Soundings :
226. Enumeration of Methods 294
227. By Two Angles read on Shore 295
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CONTENTS. xxiii
PAGB
228. By Two Angles read in the Boat— The Three-point Problem 295
229. By one Range and one Angle 298
230. Buoys, Buoy-flags, and Range-poles 299
231. By one Range and Time-intervals. . 300
232. By means of Intersecting Ranges 300
833. By Means of Cords or Wires 300
Making thb Soundings :
234. The Lead 301
235. The Line 301
236. Sounding Poles 303
237. Making Soundings in Running Water 303
238. The Water-surface Plane of Reference 303
239. Lines of Equal Depth , 304
240. Soundings on Fixed Cross-sections in Rivers 304
241. Soundings for the St jdv of Sand-waves 305
242. Areas of Cross-section 306
Bench-marks, Gauges, Water-levels, and Water-Slope :
243. Bench-marks 307
244. Water Gauges 307
245. Water-levels 308
246. River-slope 309
The Discharge of Streams :
247. Measuring Mean Velocities of Water- currents 310
248. Use of Sub-surface Floats 311
249. Use of Current Meters 316
250. Rating the Meter , 317
251. Use of Rod Floats 323
252. Comparison of Methods 324
253. The Relative Rates of Flow in Diflferent Pans of ihe Cross section 325
254. To find the Mean Velocity on the Cross-section 328
255. Sub-currents 331
256. The Flow over Weirs 332
257. Weir Formulae and Corrections 335
258. The Miner's Inch 338
259. Formulae for the Flow of Water in Open Channels— Kutter's For-
mula 339
360. Cross-sections of Least Resistance 344
ScDiMENT Observations:
261. Methods and Objects 345
362. Collecting the Specimens of Water 347
S63. Measuring out the Samples 347
264. Siphoning off, Filtering, and Weighing the Sediment. 348
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X^lV CONTENTS
CHAPTER XI.
MINING SURVEYING.
PACB
Surveying Mining Claims :
265. Title to Mining Claims 349
266. Location Surveys 351
267. Surveying Lode Claims 351
268. Patent Surveying 355
269. Placer Claims 368
270. Mill Sites 368
271. Amended Surveys 368
272. Adverse Surveys 369
Underground Surveys :
273. Underground Surveying 370
274. Instruments 370
274^. Stations 377
274^. Carrying the Meridian into the Mine 380
274^. Underground Traversing 386
274^. Underground Leveling 389
274/. Mapping the Survey 390
274/". Problems of Underground Surveying 392
274^. Surface Surveys 397
274^. Court Maps 398
CHAPTER XII.
CITY SURVEYING.'
275. Lana-surveying Methods inadequate in City Work 400
276. The Transit 401
277. The Steel Tape 401
Laying Out a Town Site :
278. Provision for Growth 403
279. Contour Maps 404
280. The Use of Angular Measurements in Subdivisions 404
281. Laying out the Ground 405
282. The Plat to be Geometrically Consistent 407
283. Monuments 407
284. Surveys for Subdivision 409
285. The Datum-plane 413
286. The Location of Streets 413
887. Sewer Systems 414
288. Water-supply 414
289. The Contour Map 415
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CONTENTS. XXV
Methods of Measurement : page
290. The Retracing of Lines 415
291. Erroneous Standards 416
292. True Standards 417
293. The Use of the Tape 418
294. Determination of the '* Normal Tension " 420
295. The Working Tension 424
296. The Effect of Wind 425
297. The Effect of Slope • 426
298. The Temperature Correction 426
299. Checks 427
Miscellaneous Problems :
300. The Improvement of Streets 428
301. Permanent Bench-marks 428
302. The Value of an Existing Monument 429
303. The Significance of Possession 431
304. Disturbed Corners and Inconsistent Plats 432
305. Treatment of Surplus and Deficiency 433
306. The investigation and Interpretation of Deeds 435
307. Office Records 435
308. Preservation of Lines 436
309. The Want of Agreement between Surveyors 437
CHAPTER XIIL
THE MEASURMEENT OF VOLUMES.
310. Proposition 438
311. Grading over Extended Surfaces 440
312. Approximate Estimates by Means of Contours 443
313. The Prismoid 448
314. The Prismoidal Formula 448
315. Areas of Cross-section 450
316. The Centre and Side Heights 451
317. The Area of a^Three-level Section 451
318. Cross-sectioning 452
319. Three-level Sections, the Upper Surface consisting of two
Warped Surfaces 454
320. Construction of Tables for Prismoidal Computation 456
321. Three-level Sections, the Surface divided into Four Planes by
Diagonals 461
322. Comparison of Volumes by Diagonals and by Warped Surfaces 463
323. Preliminary Estimates from the Profiles 465
324. Borrow- pits 468
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XXVI CONTENTS.
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325. Shrinkage of Earthwork 468
326' Excavations under Water 469
CHAPTER XIV.
GEODETIC SURVEYING.
327. Objects of a Geodetic Survey 472
328. Triangulation Systems 473
329. The Base-line and its Connections ' 475
330. Tb e Reconnaissance 477
331. Instrumental Outfit for Reconnaissance 479
332. The Direction of Invisible Stations 480
333. The Heights of Stations 480
334. Construction of Stations 485
335- Targets 486
336. Heliotropes 490
337. Station Marks 492
Measurement of the Base Line:
338. Methods 495
The Steel Tape 497
339. Method of Mounting and Stretching the Tape 498
340. M. jaderin's Method 501
341. The Absolute Length of Tape 503
342. The Coefficient of Expansion 504
343. The Modulus of Elasticity 505
344. Effect of the Sag 505
345. Temperature Correction 507
346. Temperature Correction when a Metallic Thermometer is used 508
347. Correction for Alignment 510
348. Correction for Sag '. 513
349. Correction for Pull 513
350. Elimination of Corrections for Sag and Pull 513
351. To reduce a Broken Base to a Straight Line 516
352. To reduce the Length of the Base to Sea-level 516
353. Summary of Corrections 517
354. To compute any Portion of a Broken Base which cannot be
directly measured 520
355. Accuracy attainable by Steel-tape and Metallic-wire Measure-
ments 52X
Measurement of the Angles :
356. The Instruments 525
357. The Filar Micrometer 528
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CONTENTS, XXVI 1
PACE
358. The Programme of Observations 531
359. The Repeating Method 532
360. Method by Continuous Reading around the Horizon 533
361. Atmospheric Conditions 535
362. Geodetic Night Signals 536
363. Reduction to the Centre 536
Adjustment of the Measured Angles :
364. Equations of Condition 539
365. Adjustment of a Triangle 541
Adjustment of a Quadrilateral :
366. The Geometrical Conditions 542
367. The Angle-equation Adjustment 542
368. The Side-equation Adjustment 545
369. Rigorous Adjustment for Angle- and Side-equations 549
Example of Quadrilateral Adjustment 552
Adjustment of Larger Systems:
370. Used only in Primary Triangulation 554
371. Computing the Sides of the Triangles 554
Latitude and Azimuth:
372. Conditions 558
373. Latitude and Azimuth by Observations on Circumpolar Stars
at Culmination and Elongation 558
374. The Observation for Latitude 562
375. First Method 563
376. Second Method 563
377. Correction for Observations not on the Meridian 564
378. The Observation for Azimuth 565
379. Corrections for Observations near Elongation 566
380. The Target 568
381. The Illumination of Cross- wires 568
381a. Azimuth from Polaris at any Hour 569
Time and Longitude:
382. Fundamental Relations 571
383. Time 572
384. Conversion of a Sidereal into a Mean Solar Time Interval, and
vice versa 575
385. To change Mean Time into Sidereal Time 576
386. To change from Sidereal to Mean Time 577
387. The Observation for Time 578
388. Selection of Stars, with List of Southern Time-Stars for each
Month 578
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XXVUi CONTEN-TS,
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389. Finding the Mean Time by Transit 582
390. Finding the Altitude 583
391. Making the Observations 584
392 Longitude 586
393. Computing the Geodetic Positions 587
394. Exam pie oi L M Z Computation 591
Geodetic Levelling:
395. Of Two Kinds 592
{A) Trigonometrical Levelling:
396. Refraction 592
397. Formulae for Reciprocal Observations 593
398. Formulae for Observations at One Station only 595
399. Formulae for an Observed Angle of Depression to a Sea Horizon 597
400. To find the Value of the Coefficient of Refraction 598
{B) Precise Spirit-Levelling:
401. Precise Levelling Defined 599
402. The Instruments 600
403. The Instrumental Constants 603
404. The Daily Adjustments 607
405. Field Methods 609
406. Limits of Error 612
407. Adjustment of Polygonal Systems 613
408. Determination of the Elevation of Mean Tide 617
CHAPTER XV.
PROJECTION OF MAPS, MAP-LETTERING, AND TOPOGRAPHICAL SYMBOLS.
Projection of Maps:
409. Purpose of the Map 6x8
410. Rectangular Projection 618
411. Trapezoidal Projection..... 619
412. The Simple Conic Projection 620
413. De r Isle's Conic Projection 621
414. Bonne's Projection 621
415. The Polyconic Projection 622
416. Formulae used in the Projection of Maps 622
417. Meridian Distances in Table VIII 625
418. Summary 626
419. The Angle of Convergence of Meridians 628
Map-Lettering and Topographical Symbols:
420. Map-Lettering 629
421. Topographical Symbols 630
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CONTENTS. XxJx
APPENDIX A.
The Judicial Functions of Surveyors 633
APPENDIX B.
Instructions to U. S. Deputy Mineral Surveyors 643
APPENDIX C.
Finite Differences 685
APPENDIX 3).
Derivation of Geodetic Formula 691
APPENDIX £.
Gf/kjraphical Positions of Base Lines and Principal Meridians Gov-
erning the U. S. Land Surveys 702
APPENDIX F.
Instructions for Secondary Triangulation, Precise Level, and
Topographical and Hydrographical Field Work under
the Mississippi River Commission, 1891 7^5
APPENDIX G.
The Essential Requirements of a Survey and Map, and the Owner-
ship of Surveys 724
APPENDIX H.
AJicHiGAN Laws for Making Town, City, and Village Plats 73'
APPENDIX L
Rkstorat ion of Lost or Obliterated Corners 750
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XXX CONTENTS,
TABLES.
PAGB
I.— Trigonometrical Formula 753
II.— For Converting Metres, Feet, and Chains 757
III.— Logarithms of Numbers to Four Places 758
IIIa. — Logarithms of Trigonometrical Functions to Four Places. 760
IV. — Logarithmic Traverse Table 764
V. — Stadfa Reductions for Horizontal Distance and for Ele-
vation 772
VI. — Natural Sines and Cosines 780
VII. — Natural Tangents and Cotangents 789
VIII. — Coordinates for Polvconic Projection 801
IX. — Values of Coefficient in Kutter's Formula 802
X. — Diameters of Circular Conduits, by Kutter's Formula ... 803
XI. — Earthwork Table— Volumes by the Prismoidal Formula . . 804
XII. — Azimuths op Polaris for all Hours 814
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SURVEYING.
INTRODUCTION.
Surveying is the art of making such field observations and
measurements as are necessary to determine positions, areas,
volumes, or movements on the earth^s surface. The field opera-
tions employed to accomplish any of these ends constitute a
survey. Accompanying such survey there is usually the field
secord,the computation, and the final maps, plats, profiles, areas,
or volumes. The art of making all these belongs, therefore, to
the subject of surveying.
Inasmuch as all fixed engineering structures or works involve
a knowledge of that portion of the earth's surface on which they
arc placed, together with the necessary or resulting changes in
the same, so the execution of such works is usually accompa-
nied by the surveys necessary to obtain the required informa-
tion. Thus surveying is seen to be intimately related to en-
gineering, but it should not be confounded with it. All
engineers should have a thorough knowledge of surveying, but
a surveyor may or may not have much knowledge of engineer-
ing.
The subject of Surveying naturally divides itself into —
I. The Adjustment, Use, and Care of Instruments.
II. Methods of Field Work.
III. The Records, Computations, and Final Products.
All the ordinary instruments that a surveyor may be called
upon to use in any of the departments of the work will be dis-
cussed in the following pages. The most approved methods
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MTJ^ODUdTldl^,
only will be given for obtaining the desired information, and
many problems that are mure curious than useful will not be
mentioned. The student is a'^sumed to possess a knowledge of
geometry, and of plane and spherical trigonometry. He is also
supposed to be guided by an instructor, and have access to
most of the instruments here mentioned, with the privilege of
using them in the field.
The field work of surveying consists wholly of measuring dis-
tances, angles, and time, and it is well to remember that no meas-
urement can ever be made exactly. The first thing the young sur-
veyor needs to learn, therefore, is t\\^ proportionate error allowable
in the special work assigned him to perform. It is of the utmost
importance to his success that he shall thoroughly study this
subject. He should know what all the sources of error are, and
their relative importance ; also the relative cost of diminishing
the size of such errors. Then, with a given standard of accuracy,
he will know how to make the survey of the required standard
with the least expenditure of time and labor. He must not do
all parts of the work as accurately as possible, or even with the
same care. For, if the expense is proportioned to the accuracy
of results, then he is the most successful surveyor who does his
work just good enough for the purpose. The relative size of
the various sources of error is of the utmost importance. One
should not expend considerable time and labor to reduce the
error of measurement of a line to i in 10,000 when the unknown
error in the length of the measuring unit may be as high as i
m 1000.
The surveyor must carefully discriminate, also, between com-
pensating errors and cumulative errors. A compensating error
is one which is as likely to be plus as minus, and it is therefore
largely compensated in, or eliminated from, the result. A
cumulative error is one which always enters with the same sign,
and therefore it accumulates in the result. Thus, in chaining,
the error in setting the pin is a compensating error, while the
error from erroneous length of chain is a cumulative error. If a
mile is chained with a 66-foot chain, there are 80 measurements
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INTRODUCTION,
taken. Suppose the error of setting the pin be 0.5 inch, and the
error in the length of the chain be o.i inch. Now the theory of
probabilities shows us that in the case of compensating errors
the square root of the number of errors probably^ remains un-
compensated. The probable error from setting the pins is
therefore 9 X 0.5 inch = 4.5 inche*!. The error from erroneous
length of chain is 80 X o.i inch ^ 8 inches. Thus we see that
although the error from setting the pins was five times a« great as
that from erroneous length of chain, yet in running on% mile, the
resulting error from the latter cause was nearly twice that from
the former. A careful study of the various sources of error
afifecting a given kind of work will usually enable the surveyor
either to add greatly to its accuracy without increasing its cost,
or to greatly diminish its cost without diminishing its accuracy.
The surveyor should have no desire except to arrive at the
truth. This is the true scientific spirit. He should be most
severely honest with himself. He should not allow himself
to change or " fudge" his notes without sufficient warrant,
and then a full explanation should be made in his note-book.
Neither should he make his results appear more accurate than
they really are. He should always know what was about the
relative accuracy with which his field work was done, and carry
his results only so far as the accuracy of the work would war-
rant. He is either foolish or dishonest who, having made a
survey of an area, for instance, with an error of closure of i in
300, should carry his results to six significant figures, thus giv-
ing the area to i in 500,000. It is usual to carry the computa-
tions one place farther than the results are known, in order that
no additional error may come in from the computation. It is
not unusual, however, to see results given in published docu-
ments to two, three, or even four places farther than the observa-
tions would warrant.
*The meaning of this statement is that on tht average this will occur oftener
than any other combitfation, and that any single result will, on the average^ be
nearer to this result than to any other.
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INTRODUCTION.
The student should make himself familiar with the structure
and use of every part of every instrument put into his hands.
The best way of doing this is to take the instrument all apart
and put it together again. This, of course, is not practicable for
each student in college, but when he is given an instrument in
real practice, he should then make himself thoroughly familiar
with it before attempting to use it.
The adjustments of instruments should be studied as problems
in descriptive geometry and not as mechanical manipulations,
learned in a mechanical way; and when adjusting an instrument
the geometry of the problem should be in the mind rather than
the rule in the memory.
Students of engineering in technical schools are urged to
make themselves familiar with every kind of instrument in the
outfit of the institution, and to do in the field every kind of work
herein described if possible. Otherwise he may be called upon
to do, or to direct others to do, what he has never done himself,
and he will then find that his studies prove of little avail with-
out the real knowledge that comes only from experience.
smith's field manual.*
Professor Leonard S. Smith, of the University of Wisconsin,
has prepared a field manual of notes and problems for the use
of students in surveying, especially designed to accompany
this work. Teachers and students using this work as a text-
book will find the Manual of Prof. Smith very helpful. It con-
tains fifty-five problems, and while a number of them have
reference to particular marks on the campus of the University
of Wisconsin and in the city of Madison, these could be readily
changed to suit any set of local conditions. This manual is
neatly bound in red morocco and can be obtained from the
author.
* This note added in the sixteenth and subsei^nt editions.
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BOOK I.
ADJUSTMENT, USE, AND CARE OF INSTRUMENTS.
/
CHAPTER I.
INSTRUMENTS FOR MEASURING DISTANCES.
THE CHAIN.
and
1. The Eng^ineer's Chain is 50 or 100 feet
should be made of No. 12 steel wire. The
links are one foot long, including the con-
necting ring?. All joints in rings and links
should be brazed to prevent giving. The
connections are designed so as to admit of
as little stretch as possible. Every t^nth
foot is marked by a special form of brass
tag. If the chain is adjustable in length,
it should be made of standard length by
measuring from the inside of the handle
at one end to the outside ^f the handle at
the other. If it is not adjuHjtable, measure
from the outside of the hanjle at the rear
end to the standard mark at the forward
end.
2. Gunter's Chain is 66 i..^ j^^^^ ^^^ ^^ divided into 100
links, each link being 7.92 incht .^ length. This chain is
mostly used in land-surveying, wht^v.^^^^^^.
Fig.
measure.
It was invented by Edmund ^^unier
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SURVEYING,
astronomer, about 1620, and is very convenient for obtaining
areas in acres or distances in miles. Thus,
One mile = 80 chains ; also,
One acre = 160 square rods,
= 10 square chains,
= ioo,ocx> square links.
If, therefore, the unit of measure be chains and hundredths
(links), the area is obtained in square chains and decimals, and
by pointing off one more place the result is obtained in acres.
This is the length of chain used on all the U. S. land surveys.
In all deeds of conveyance and other documents, when the
word chain is used it is Gunter*s chain that is meant.
3. Testing the Chain. — No chain, of whatever material
or manufacture, will remain of constant length. The length
changes from temperature, wear, and various kinds of distor-
tion. A change of temperature of 70° F. in a 100-foot chain
will change its length by 0.05 foot, or a change of i in 2000.
If the links of a chain are joined by three rings, then there
are eight wearing surfaces for each link, or eight hundred
wearing surfaces for a 66- or 100-foot chain. If each surface
should wear 0.01 inch, the chain is lengthened by eight inches.
It is not uncommon for a railroad survey of, say, 300 miles to
be run with a single chain. If such a chain were of exactly the
right length at the beginning of the survey, it might be six
inches too long at the end of it.
The change of length from distortion may come from a
flattening out of the connecting rings, from bending the links,
or from stretching the chain beyond its elastic Hmit, thus giv-
ing it a permanent set. Both the"V««J^ and the distortion are
likely to be less for a steel chain t/^n for an iron one. When
a bent link is straightened it i*? permanently lengthened.
When we remember \>^'^ ^^1 unknown changes in the
length of the chain pr-'^^ce cumulative errors in the meas-
~ •'.low important it is that the true length of
V
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ADJUSTMENT, VsE, Al^t) CAkE OP INSTRUMENTS. 7
the chain should be always known, or better, that the standard
length (50, 66, or 100 feet) should be properly measured from
one end of the chain and marked at the other. This chain
test is most readily accomplished by the aid of a standard steel
tape, which is at least as long as the chain. By the aid of such
a tape a standard length may be laid off on the floor of a large
room, or two stones may be firmly set in the ground at the
proper distance apart and marks cut upon their upper sur-
faces. If stones are used they should reach below the frost-
line. Or a short tape, or other standard measuring unit, may
be used for laying off such a base-line. By whatever means it
is accomplished, some ready means should at all times be
available for testing the chain. Since a chain always grows
longer with use, the forward end of the chain will move
farther and farther from the standard mark. A small file-
mark may be made on the Jiandle or elsewhere, and then re-
moved when a new test giv 1& a new position. Care must be
exercised to see that there are no kinks in the chain either in
ti
testing or m use.
In laying (^^t the standard base the temperature at which
the unit of measure is standard should be known (this tempera-
ture is stamped on the better class of steel tapes), and if the
base is not laid out at this temperature, a correction should
be made before the marks are set. The coefficient of expansion
of iron and steel is very nearly 0.0000065 for 1° F. If T^ be
the temperature at which the tape is standard, T the tem-
perature at which the base is measured, and L the length
of the base, then 0.0000065 (T^— T)L is the correction to be
applied to the measured length to give the true length.
When the chain is tested by this standard base the tem-
perature should be again noted, and if this is about the mean
temperature for the field measurements no correction need be
made to the field work. If it is known, at the time the chain
is tested, that the temperature is very different from the prob-
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8 SURV^EYING.
able mean of the field work, then the standard mark can be so
placed on the chain as to make it standard when in use.
4. The Use of the Chain. — The chain is folded by taking
it by the middle joint and folding the two ends simultaneous-
ly. It is opened by taking the two handles in one hand and
throwing the chain out with the other.
Since horizontal distances are always desired in surveying,
the chain should be held horizontally in measuring. Points
vertically below the ends of the chain are marked by iron pins,
the head chainman placing them and the rear chainman remov-
ing them after the next pin is set. The chain is lined in either
by the head or rear chainman, or by the observer at the instru-
ment, according as the range-pole is in the rear, or in front, or
not visible by either chainman. When chaining on level
ground, the rear chainman brings the outside of the handle
against the pin, and the head chainman sets the forward side
of his pin even with the standard ftiark on the chain. By this
means the centres of the pins are the tru^ iistance apart. On
uneven ground both chainmen cannot hoTd^to the pin ; one end
being elevated in order to bring the chain <jo a horizontal
position. In this case there are three difficulties to be over-
come. The chain should be drawn so taut that the stretch
from the pull would balance the shortening from the sag; the
chain should be made horizontal ; the elevated end-mark must
be transferred vertically to the ground. It is practically im-
possible to do any of these exactly. The first could be deter-
mined by trial. Stretch the chain between two points at the
same elevation, having it supported its entire length. Then
remove the supports, and see how strong a pull is required to
bring it to the marks again. This should be done by the chain-
men themselves, thus enabling them to judge how hard to pull
it when it is off the ground. To hold the chain horizontal on
sloping ground is very difficult, on account of the judgment
being usually very much in error as to the position of a hori-
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS,
jontal line. In all such cases the apparently horizontal line is
much too nearly parallel with the ground. Sometimes a level has
been attached to one end of the chain, in which case it should
be adjusted to indicate horizontal end-positions for a certain
pull, this being the pull necessary to overcome the shortening
from sag. To hold a plumb-line at the proper mark, with the
chain at the right elevation, and stretched the proper amount,
requires a steady hand in order that the plumb-bob may hang
stationary. This should be near the ground, and when all is
ready, it is dropped by the chainman letting go the string.
The pin is then stuck and the work proceeds. It is common
in this country for the rear chainman to call *' stick*' when he
is ready, and for the head chainman to answer "stuck*' when
he has set the pin. The rear cRainman then pulls his pin and
walks on.
There should be eleven pins, marked with strips of colored
flannel tied in the rings to assist in finding them in grass or
brush. In starting, the rear chainman takes a pin for the initial
point, leaving the head chainman with ten pins. When the
last pin is stuck, the head chainman calls "out," and waits by
this station until the rear chainman comes up and delivers over
the ten pins now in his possession. The eleventh pin is in the
ground, and serves as the initial point for the second score.
Thus only every ten chains need be scored.
Good chaining, therefore, consists in knowing the length of
the chain, in true alignment, horizontal and vertical, and in
proper stretching, marking, and scoring.
THE STEEL TAPE.
5. Varieties. — Steel tapes are now made from one yard to
1000 feet in length, graduated metrically, or in feet and tenths.
A pocket steel tape from three to ten feet long should always
be carried by the surveyor. A 50-foot tape is best fitted to
city surveying where there are appreciable grades. For cities
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lO SURVEYING.
without grades a ic»foot tape might be found more useful.
For measuring base-lines, or for some kinds of mining surveying,
a 300 or 5CX) foot tape is best. These are of small cross-section,
being about oj inch wide and 0.02 inch thick. A tape about
Fig. 3.
0.5 inch wide and 0.02 fnch thick (Fig. 2) is perhaps best suited
to general surveying.
6. The Use of Steel Tapes. — Steel tape-measures are used
just as chains are. They are provided with handles, but the
end graduation-marks are usually on the tape itself and not on
the handle. They are graduated to order, the graduations
being either etched or made on brass sleeves which are fastened
on the tape. Their advantages are many. They do not kink,
stretch, or wear so as to change their length, so that, with
careful handling, they remain of constant length except for
temperature. They are used almost exclusively in city and
bridge work, and in the measurement of secondary base-lines.
The same precautions must be taken in regard to alignment,
pull, and marking with the tape, as was described for the
chain.*
* For methods of using the steel tape in accurate measurements, sec Chap-
ter XIV., Base-Line Measurements.
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ADJVsrMEiSrT, VsE, AMD CAkE OP tMsTkVMkM% tt
EXERCISES.
To be vtorked out on the ground by the use of the chain or tape alone,
7. To chain a line over a hill between two given points, not visible from
each other.
Range-poles are set at the given points. Then the two chainmen, each with
a range-pole, range themselves in between the two fixed points, near the top
of the hill, by successive approximations. The line can then be chained.
8. To chain a line across a valley between two fixed points.
Establish other range-poles by means of a plumb-line held on range between
the points.
9. To chain a line between two fixed points when woods intervene, and the
true line is not to be cleared out.
Range out a trial line by poles, leaving fixed points. Find the resulting error
at the terminus, and move all the points over their proportionate amount. The
true line may then be chained.
10. To set a stake in a line perpendicular to a given line at a given point.
All multiples of 3, 4, and 5 are the sides of a right-angled triangle; also any
angle in a semicircumference is a right angle.
11. To find where a perpendicular from a given point without a line will meet
that line.
Run an inclined line from the given point to the given line. Erect a per-
pendicular from the given line near the required point, extend it till it intersects
ihe inclined line, and solve by similar triangles.
12. To establish a second point that shall make with a given point a line
parallel to a given line.
Diagonals of a parallelogram bisect each other.
13. To determine the horizontal distance from a given point to a visible but
Inaccessible object.
Use two similar right-angled triangles.
14. To prolong a line beyond an obstaclef in azimuth* and distance.
First Solution : By an equilateral triangle.
Second Solution : By two rectangular offsets on each side of the obstacle.
Third Solution : By similar triangles, as in Fig. 3.
From any point as A run the line AB, fixing the half and three quarter points
at X and y. From any other point as C, run CxD^ making xD = Cx. From D
*The azimuth of a line is the angle it forms with the meridian, and is meas-
ured from the south point in the direction S.W. N.E. to 360 degrees. It thus
becomes a definite direction when the angle alone is given. Thus the azimuth
of 220'' corresponds to the com pass- bearing of N. 40** £.
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12
SURVEYING.
run DyE making DE = 4 Z>y, fixing the middle point 2. From B run BzH^
making zH = Bs. Then is HE parallel and equal to DB^ AC, and CII
A C I r ^ E
^^--...^^^ I I H^
D B
Fig. 3.
Check: Measure
Stakes should be set at all the points lettered in the figure.
HE and AC, If they are equal the work is correct.
15. To measure a given angle.
Lay oflf equal, distances, b, from the vertex on the twolines, and measure the
a
third side a of the triangle. Then X2Si\ A= —-=f=
r 4^« — a*
16. To lay out a given angle on the ground.
Reverse the above operation. A is known; assume b and compute a. Then
from A measure oS AB ■= b. From B and A strike arcs with radii equal to a
and b respectively, giving an intersection at C. Then CAB is the required
angle. If b is assumed not greater than 0.6 the length of the chain, angles may
be laid out up to 90*.
~:^<>
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._.«._.. oc
17. To mark a point on a house, bowlder, or
other object, near a line of survey, which shall
be at right angles to a given point in said line,
by means of a chain tape, or cord.
Let AB he 3L line of survey. Let C be a
house on which a transfer of point B is re-
quired at right angles to AB, Then with
radius BA swing arc AO, and with \ BC on
stick or tape measure to arc from line AB 2X
point O ; with radius OB (or BA)^ from 0
swing arc touching house in C as required.
Fig. 30.
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ADJUSTMENT^ USE, AND CARE OF INSTKUMEXTS. I3
CHAPTER II.
INSTRUMENTS FOR DETERMINING DIRECTIONS.
THE COMPASS.
18. The Surveyor's Compass consists essentially of a line
of sight attached to a horizontal graduated circle, at the centre
of which is suspended a magnetic needle free to move, the
whole conveniently supported with devices for levelling. Fig,
Fig. 4.
4 shoAVs a very good form of such an instrument. In ad-
dition to the above essential features, the instrument here
shown has a tangent-screw and vernier-scale at e for setting
off the declination of the needle; a tangent-scale on the edge
of the vertical sight for reading vertical angles, the eye being
placed at the sight-disk shown on the opposite standard; and an
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J4 ^Uki^EViMti,
auxiliary graduated circle, with vernier, shown on the front
part of the plate, for reading angles closer than could be done
with the needle. The compass is mounted either on a tripod
or on a single support called a Jacob's-staff. It is connected
to its support by a ball-and-socket joint, which furnishes a con-
venient means of levelling.
Although the needle-compass does not give very accurate
results, it is one of the most useful of surveying instruments.
Its great utility lies in the fact that the needle always points
in a known direction, and therefore the direction of any line
of sight may be determined by referring it to the needle-bear-
ing. The needle points north in only a few localities ; but its
declination from the north point is readily determined for any
region, and then the true azimuth, or bearing of a line, may be
found. It has grown to be the universal custom, in finding
the direction of a line by the compass, to refer it to cither
the north or the south point, according to which one gives an
acute angle. Thus, if the bearing is icx)° from the south
point it is but 8o° from the north point, and the direction
would be defined as north, 80° east or west, as the case
might be: thus no Hne can have a numerical bearing of
more than 90®. In accordance with this custom, all needle-
compasses are graduated from both north and south points
each way to the east and west points, the north and south
points being marked zero, and the east and west points 90°.
When the direction of a line is given by this system it is
called the bearing of the line. When it is simply referred
to the position of the needle it is called the magnetic bearing.
When it is corrected for the declination of the needle,
either by setting off the declination on the declination-arc or
by correcting the observed reading, it is called the true bear-
ing, being then referred to the true meridian.
Because the graduated circle is attached to the line of sight
and moves with it, while the needle remains stationary, E and
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS fj
W are placed on the compass-circle in reversed position.
Thus when the line of sight is north-east, the north end of the
needle points to the left of the north point on the circle, and
hence E must be put on this side of the meridian line.
In reading the compass,, always keep the north end of the circle
pointing forward along the line, and read the north end of the
needle.
The north end of the needle is usually shaped to a special
design, or, if not, it may be distinguished by knowing that the
south end is weighted by having a small adjustable brass wire
slipped upon it to overcome the tendency the north end has
to dip.
ADJUSTMENTS OF THE COMPASS.
19, The General Principle of almost all instrumental ad-
justments is the Principle of Reversion, whereby the error is
doubled and at the satne time made apparent, A thorough mas-
tery of this principle will nearly always enable one to deter-
mine the proper method of adjusting all parts of any survey,
ing instrument. It should be a recognized principle in sur-
veying, that no one is competent to handle any instrument
who is not able to determine when it is in exact adjustment,
to locate the source of the error if not in adjustment, to dis-
cuss the effect of any error of adjustment on the work in
hand, and to properly adjust all the movable parts. The
methods of adjustment should not be committed to memory —
any more than should the demonstration of a proposition in
geometry. The student in reading the methods of adjust-
ment should see that they are correct, just as he sees the cor-
rectness of a geometrical demonstration. Having thus had
the method and the reason therefor clearly in the mind, he
should trust his ability to evolve it again whenever called
upon. He thus relies upon the accuracy of his reasoning,
rather than on the distinctness of his recollection,
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l6 SURVEYING.
20. To make the Plate perpendicular to the Axis of the
Socket. — This must be done by the maker. It is here men-
tioned because the axis is so likely to get accidentally bent.
Instruments made of soft brass must be handled very care-
fully to prevent such an accident. If this adjustment is found
to be very much out, it should be sent to the makers. If
much out, it will be shown by the needle, and also by the
plate-bubbles.
21. To make the Plane of the Bubbles perpendicular
to the Axis of the Socket. — Level it in one position, turn
1 80°, and correct one half the movement of each bubble by
the adjusting-screw at the end of the bubble-case. Now level
up again, and revolve 180°, and the bubbles should remain at
the centre. If not, adjust for one half the movement again,
and so continue until the bubbles remain in the centre for all
positions of the plate.
The student should construct a figure to illustrate this and almost all other
adjustments. Thus, in this case, let the figure consist of two lines, one repre-
senting the axis of the socket, and the other the axis of the bubble, crossing it.
Now if these two lines are not at right angles to each other, when the one is
horizontal (as the bubble-axis is when the bubble rests at the centre of its tube)
the other is inclined from the vertical. Now with this latter fixed, let the
figure be revolved 180'' about it (or construct another figure representing such
a movement), and it will be seen that the bubble-axis now deviates from the
horizontal by ttoice the difference between the angle of the lines and 90*". By
now correcting one half of this change of direction on the part of the bubble-
axis, it will be made perpendicular to the socket-axis. Then by relevelling the
instrument, which consists of moving the socket-axis until the bubbles return
to the middle of the tubes, the Instrument should now revolve in a horizontal
plane.
22. To adjust the Pivot to the Centre of the Graduated
Circle. — When the two ends of the needle do not read exactly
alike it may be due to one or more of three causes: The
circle may not be uniformly graduated ; the pivot may be bent
out of its central position ; pr the needle may be bent. All
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ADJUSTMENT, USE, AND CAKE OF' INSTRUMENTS, I>
our modern instruments are graduated by machiner}'', so that
they have no errors of graduation that could be detected by
eye. One or both of the other two causes must therefore ex-
ist. If the difference between the two end-readings is con-
stant for all positions of the needle, then the pivot is in the
centre of the circle, but the needle is bent. If the difference
between the two end-readings is variable for different parts of
the circle, then the pivot is bent, and the needle may or may
not be straight. To adjust the pivot, therefore, find the posi-
tion of the needle which gives the maximum difference of end-
readings, remove the needle, and bend the pivot at right angles
to this position by one half the difference in the extreme variation
of end-readings. Repeat the test, etc. Since the glass cover
is removed from the compass-box in making this adjustment,
it should be made indoors, to prevent any disturbance from
wind.
23. To straighten the Needle, set the north end exactly
at some graduation-mark, and read the south end. If not 180^
apart, bend the needle until they are. This implies that the
preceding adjustment has been made, or examined and found
correct.
24. To make the Plane of the Sights normal to the
Plane of the Bubbles. — Carefully level the instrument and
bring the plane of the sights upon a suspended plumb-line.
If this seems to traverse the farther slit, then that sight is in
adjustment. Reverse the compass, and test the other sight
in like manner. If either be in error, its base must be re-
shaped to make it vertical.
25. To make the Diameter through the Zero-gradua*
tions lie in the Plane of the Sights. — This should be done by
the maker, but it can be tested by stretching two fine hairs
vertically in the centres of the slits. The two hairs and the
two zero-graduations should then be seen to lie in the same
plane. The de$Jination-arc must be set to read zero.
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1 8 SURVEYING.
26. To remagnetize the Needle. — Needles sometimes lose
their magnetic properties. They must then be remagnetized.
To do this take a simple bar-magnet and rub each end of the
needle, from centre towards the ends, with the end of the
magnet which attracts in each case. In returning the magnet
for the next stroke lift it up a foot or so to remove it from
the immediate magnetic field, otherwise it would tend to nul-
lify its own action. The needle should be removed from the '
pivot in this operation, and the work continued until it shows
due activity when suspended. An apparently sluggish needle
may be due to a blunt pivot. If so, this should be removed,
and ground down on an oil-stone.
THE VERNIER.
27. The Vernier is an auxiliary scale used for reading frac-
tional parts of the divisions on the main graduated scale or limb.
H we wish to read to tenths of one division on the limb, we
make 10 divisions on the vernier correspond to either 9 or ii
divisions on the limb. Then each division on the vernier is
one tenth less or greater than a division on the limb. If we
wish to read to twentieths or thirtieths of one division on the
'.imb, there must be twenty or thirty divisions on the vernier
corresponding to one more or less on the limb.
The zero of the vernier scale marks the point on the limb
whose reading is desired.
Suppose this zero-point corresponds exactly with a division
on the limb. The reading is then made wholly on the limb.
If a division on the vernier is less than a division on the limb,
then, by moving the warmer forwatd a trifle, the next forward
division on the vernier corresponds with a division on the limb.
(The particular division on the limi that may be in coincidence
is of no consequence.) On the other hand, if a division on the
vernier \s greater than a division on the limb, then by moving
the vernier forward a trifle, the next backward division on the
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AiyJVsTME^t, UsK AS'b CAkk dp" iMsr/^t/MEArrs. 19
vernier comes into coincidence. Thus we have two kinds of
verniers, called dirfct and retrosrade according as they are read
forward or backward from the zero-point. Most verniers in
use are of the direct kind, but those commonly found on sur-
veyors' compasses for setting off the declination are generally
of the retrograde order.
10 u
Pig.
In Fig. 5 are shown two direct verniers, such as are used
on transits with double graduations. Thus in reading to the
right the reading is 138^45', but in reading to the left it is 221° .
15'. In each case we look along the vernier in the direction of
the graduation for the coincident lines.
Fig. 6.
In Fig. 6 is shown a special form of retrograde vernier in
which the sattie set of graduation-lines on the vernier serve for
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20 SURVEYING.
either right- or left-hand angles. Here a division of the verniei
is larger than a division on the limb, and it must therefore be
read backwards. Thus, we see that the zero of the vernier
is to the left of the zero of the Umb, the angle being 30' and
something more. Starting now toward the right (backwards)
on the vernier scale, we reach the end or 15-minute mark,
without finding coincident lines ; we then skip to the left-hand
side of the vernier scale and proceed towards the right again
until we find coincident lines at the twenty-sixth mark. The
reading is therefore 30-1-26=56 minutes. This is the form
of vernier usually found on surveyors* compasses for setting
off the declination. We have therefore the following
Rules.
First, To find the " smallest reading' of the vernier ^ divide
the value of a division on tlie limb by the number of divisions in
the vernier.
Second, Read forward along the limb to the last graduation
preceding the zero of the vernier ; then read forward along the
vernier if direct, or backward if retrograde, until coificident lines
are found. The number of this line on the vernier from the zero-
graduation is the number of ^^ smallest-reading'* units to be
added to the reading made on the Itmb.
These rules apply to all verniers, whether linear or circular
THE DECLINATION OF THE NEEDLE.
28 The Declination* of the Needle is the horizontal
angle it makes with the true meridian. At no place on the
earth is this angle a constant. The change in this angle is
called the variation of the declination.
29. The Daily Variation in the Declination consists in a
* Formerly called variation of ihe needle, and still so called by navigators
and by many surveyors.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 21
swinging of the needle through an arc of about eight minutes
daily, the north end having its extreme easterly variation about
8 A. M. and its extreme westerly position about 1.30 P.M. It
has its ffuan or trtte declination about 10.30 A.M. and 8 P.M.
It varies with the latitude and with the season, but the follow-
ing table gives a fair average for the United States. A more
extended table may be found in the Report of the U. S. Coast
and Geodetic Survey for 1881, Appendix 8.
TABLE OF CORRECTIONS TO REDUCE OBSERVED BEARINGS
TO THE DAILY MEAN.
MoifTH.
Add 10 N.E. and S W.
bearings.
Subtract from N.W. and
S.E. bearings.
Add to N.W. and S.E. bearings.
Subtract from N.E. and S.W. bearings.
6
A.M.
7 8
A.M. A.M.
9
A.M.
10
A M.
11
A.M.
12
M.
1
P.M.
a
P.M
3
P M
4
P.M.
5
P.M
6
P.M.
January
,/
1'
a'
a'
\'
0'
a'
3'
3'
a'
\'
1'
0'
April
3
4
4
3
Z
I
4
5
5
4
3
2
I
July
4
I
5
2
5
a
4
a
1
I
4
3
5
3
5
3
4
a
3
_
1 Ortobcr
X
I
0
0
This table is correct to the nearest minute for Philadelphia, where the observations were
made.
30. The Secular Variation of the magnetic declination is
probably of a periodic character, requiring two or three cen-
turies to complete a cycle. The most extensive set of obser-
vations bearing on this subject have been made at Paris, where
records of the magnetic declination have been kept for about
three and a half centuries. The secular variation for Paris is
shown in Fig. 7, and that for Baltimore, Md., in Fig. 8.*
Whether or not either of these curves will return in time to
the same extreme limits here given is unknown, as is also the
cause of these remarkable changes. The extraordinary varia-
tion in the declination at Paris of some 32°, and that at
♦ These taken from the Coast Survey Report of 1882.
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n
SUkVMYWG.
Baltimore of some 5°, show the necessity of paying careful
attention to this matter. No reliance should be placed on
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Fig. 8.
old determinations of the declination unless the rate of change
be known, and even then this rate is not likely to be constant
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 2%
a great many years. They also show the necessity of record-
ing the date and the declination of the needle on all plats and
records of surveys, with a note stating whether the bearings
given were the true or magnetic bearings at the time they
were taken.
31. Isogonic Lines are imaginary lines on the garth's sur-
face joining points whose declinations are equal at any given
time. The isogonic line joining points having no declination
is called the agonic line. There is such a line crossing the
United States passing just east of Charleston, S. C, and just
west of Detroit, Mich. All points east of this line have a
western declination, and all points west of it have an eastern
declination. The isogonic lines for 1900 for the whole of
the United States are shown on Plate I.* It will be noted
that where the observations are most thickly distributed,
as in Missouri for instance, there the isogonic lines are most
crooked ; showing that if the declinations were accurately
known for all points of this map the isogonic hnes would be
much more irregular, and would be changed very much in
position in many places.
The isogonic lines given on this chart are all moving west-
ftrard, so that all western declinations are increasing and all
eastern declinations are decreasing. They are not all moving at
the same rate, however, those in New Brunswick and also those
near the eastern boundaries of California and Oregon being
about stationary. For many points in the United States and
Canada the rate of change in the declination has been observed,
and formulae determined for computing the declination for each
point, which formulae will probably remain good for the next
twenty years. The following tables t give this information. In
these tables / is the time in calendar years. Thus for July i,
1885, /= 1885.5. In the first table all the formulae have been re-
* Reduced from the U. S. Coast and Geodetic Survey Charts,
t Taken from the U. S. C. and G. Survey Report for 1886,
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24 SURVEYING,
ferred to one date — Jan. I, 1850. Here m is used to represent
the time in years after 1850, or w= / — 1 850. Thus, for July I,
1885,;;/ = 35.5.
It will be seen that the change in the declination over the
Northern States will average about one minute to the mile in an
east and west direction. A value of the declination found in
one end of a county may be somt forty minutes in error in the
other end of the same county. This shows that the declina-
tion must be known for the exact locality of the survey. In
fact, the surveyor can never be sure of his declination until he
has observed it for himself for the given time and place. This
is best done by means of a transit instrument, and such a
method is given in the chapter on Geodetic Surveying. If,
however, no transit is at hand, a result sufficiently accurate for
compass surveying may be obtained by the compass itself.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 2$
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28
SURVEYING,
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ANNUAL CHANGE OF THE MAGNETIC DECLINATION. 2Sa
TABLE OF THE ANNUAL CHANGE of the magnetic
DECLINATION AT A NUMBER OF PRINCIPAL PLACES IN
THE United States (or close to its borders), for the
EPOCHS 1895 and 1900; extracted from Appendix No, i —
Coast and Geodetic Survey Report for 1895.*
A 4- sign indicates increasing western declination, or, what
is the same thing, decreasing eastern declination, and it will
be seen from the isogonic chart (Plate I.) for the year 1900
that the zero or agonic line passes over the eastern part of
Lake Superior, thence between the Strait of Mackinaw and
Beaver Island, centrally through Michigan, over western Ohio
and eastern Kentucky ; thence the line passes over North and
South Carolina, intersecting the coast a little below Charles-
ton ; further on it skirts the Bahama Islands. The whole line
is at present moving westward. To the right of the line we
have west, to the left of it east declination.
♦ This extract from an advanced copy of the Report was furnished through the
couitcsy of the Superintendent of the Survey.
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2Sd
SURVEYING,
I. EASTERN GROUP.
LOCALITY.
ANNUAL CHANGE IN
1895
1000
1
- 50
-5.8
-0.5?
- I.O?
-2.3
-3.1
+ 4.5
+ 3.9
+ 0.2
-0.3
+ 1.5
+ 0.9
— O.I
- 0.5
+ 2.4
+ 2.0
+ 3.0
+ 2.4
+ 2.2
+ 1-7
+ 3-8
+ 3-3
+ 2.5
+ 2.0
+ 07
+ 0.5
+ 2.1
+ 1.7
+ 3.0
+ 2.6
+ 3-4
+ 31
+ 3.0
+ 2.4
+ 3.4
+ 3.0
+ 1.6
+ 1.2
+ 1.8
+ 1.4
+ 1.6
+ I.I
+ 3.0
+ 2.0
+ 3.0
+ 2.7
+ 3.3
+ 30
+ 1.8
+ 14
+ 2.3
+ 2.0
+ 3-8
+ 3.4
+ 4.0
+ 3.7
+ 3.9
+ 3.'^
+ 1.8
+ 1.4
+ 3-3
+ 2.9
+ 2.0
+ 1.4
+ 3-3
+ 3.3
+ 4.4
+ 2.8
+ 4.8
+ 4.5
+ 29
+ 2.4
+ 3.0
+ 2.7
+ 2.8
+ 2.6
+ 2.7
+ 2.3
+ 3.7
+ 3-4
+ 3.2
+ 2.9
+ 2.8
+ 2.5
+ 2.3
+ 1.9
+ 3.7
+ 3.7
+ 2.5
+ 2.1
+ 3.7
+ 3.5
+ 3-9
St. John's, Newfoundland
Quebec, Canada
Charlottetown, Prince Edward Island
Montreal, Canada
Eastport, Me
Bangor, Me
Halifax, Nova Scotia
Burlington, Vt
Hanover, N. H
Portland, Me
Rutland. Vt
Portsmouth, N. H
Chesterfield, N. H
Newburyport, Mass
Williamstown, Mass
Albany, N. Y
Salem, Mass
Oxford, N. Y
Cambridge, Mass
Boston, Mass
Provincetown, Mass
Providence, R. I
^ Hartford, Conn
New Haven, Conn
Nantucket, Mass
Cold Spring Harbor, N. Y
New York City. N. Y
Bethlehem, Pa
Huntingdon, Pa
New Brunswick, N. J. .
Jamesburg, N. Y
Harrisburg. Pa
Hatboro, Pa
Philadelphia, Pa
Chambersburg, Pa
West Creek, Little Egg Harbor, N. J.
Baltimoie, Md
Cape May, N. Y
Washington, D. C
Cape Henlopen, Del
Williamsburg, Va
Cape Henry, Va
Newbern, N. C
Milledgeville. N. t
Charleston, S. C
Savannah, Ga
Femandina, Fla
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A}s^NUAL CkAt^GR dP THR MACNEtiC DMcLWaTION, 2&C
II. CENTRAL GROUP.
LOCALITY.
York Factory, British North America.
Fort Albany, British North America . .
Duluth, Minn
Saalt de Ste. Marie. Mich
Ficrrepont Manor, N. Y
Toronto, Canada
Grand Haven. Mich.
Milwaukee, Wis
Buffalo, N. Y
Ithaca, N. Y
Dunkirk, N. Y
Detroit, Mich
Kalamazoo, Mich
Ypsiianti, Mich
Erie, Pa
Chicago, III
Michigan City, Ind
Cleveland, Ohio
Omaha, Neb
Beaver, Pa
Pittsburg, Pa
Denver, Colo
Marietta, Ohio
Athens, Ohio
Cincinnati, Ohio
St. Louis, Mo
Nashville, Tenn
Florence, Ala
Mobile, Ala
Pensacola, Fla
Austin. Texas
New Otieans, La
San Antonio, Texas. . . . ;
Galveston, Texas
Key West, Fla
Habana, Cuba
Kingston, Jamaica
Bridgetown, Barbados
Panama, New Granada
ANNUAL CHANGE IN
1900
+ 13-5
+ 6.7
+ 2.3
+ 4.0
+ 3.7
+ 3.7
5-4
3.8
5.1
2.7
2.2
6.1
2.9
2.8
4.4
4.5
2.6
4.0
3 6
2.7
3.7
3.7
2.7
3.2
41
4.7
31
4 4
4-3
+ 4.3
+ 39
+ 29
+ 30
+ 2.0
+ 0.9
+ 2.3
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2%d
SURVEYING.
III. WESTERN GROUP.
LOCALITY.
ANNUAL CHANGE IN
1895
1900
Chamisso Island, Alaska
Port Clarence, Alaska
Port Etches, Constantine Harbor, Alaska
Port Mulgrave, Yakuiat J)ay, Alaska
Saint Paul, Kadiak Island, Alaska
Sitka, Alaska
Iliuliuk, Unalaska Island. Alaska
Petropaulovsk, Kamchatka
Nootka, Vancouver Island
Cape Flattery and Neah Bay, Wash
Port Townsend, Wash
Seattle, Wash
Olympia, Wash
Cape Disappointment, Wash
Wallawalla, Wash
Vancouver, Wash
Portland, Oregon
Salt Lake Ciiy, Utah
Cape Mendocino, Cal
San Francisco, Cal
Monterey, Cal
Santa Barbara, Cal
San Diego, Cal
El Paso. Texas
Cerros Island, Lower Cal., Mexico
Ascension Island, Ix)wer Cal., Mexico . .
Magdalena Bay, Lower Cal., Mexico
San Lucas. Lower Cal., Mexico
San Bias, Mexico
Mexico City, Mexico
Vera Cruz. Mexico
Acapulco, Mexico
+
4-
6.9
5 ?
3 ?
0 ?
+ 4 ?
- 2 ?
+ 3 o
+ 2-9
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- 0.7
- 0.4
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— I.I
+ 1-7
+ 0.2
-0.9
2 I
0.6
0.1
0.3
I.I
1-3
2-7
2.5
1.4
1 9
2.1
3-6
2 5
4-3
3.5
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+
+
+
+
+
+
+
+
+
+
+
+
+ 31
+ 27
00
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-07
- 0.7
+ 0.7
- 0.3
+ 2.7
O.I
0.0
1-5
1.6
31
2 9
1-7
2.2
2 5
3 9
2.7
4-4
3.8
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 2g
32. Other Variations of the Declination. — In addition
to the daily and secular changes in the declination, there are
others worthy of mention.
The annual variation is very small, being only about a half-
minute of arc from the mean position for the year. It may
therefore be neglected.
The lunar inequalities are still smaller, being only about fif.
teen seconds of arc from the mean position.
Magnetic disturbances are due to what are called magnetic
storms. They may occur at any time, and cannot be predicted.
They may last a few hours, or even several days. **The fol-
lowing table of the observed disturbances, in a bi-hourly series,
at Philadelphia, in the years 1840 to 1845, will ^ive an idea of
their relative frequency and magnitude :
Deviations from nor-
mal direction.
Number of
disturbances.
3'.i5 10 10'. 8
2189
10'. 8 to 18'. I
147
18'. I to 25'. 3
18
25'. 3 to 32'. 6
3
Beyond
0
"At Madison, Wis., where the horizontal magnetic intensity
is considerably less, very much larger deflections have been
noticed. Thus, on October 12, 1877, one of 48', and on May
28, 1877, one of 1° 24', were observed.*' *
The geometric axis of a needle may not coincide with its
magnetic axis, and hence the readings of two instruments at
the same station may differ slightly when both are in adjust-
ment. In this case the declination should be found for each
instruitient independently.
33. To Find the Declination of the Needle.— The
• From Report of the U. S. Coast and Geodeiic Survey lor 1882.
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iO SURVEYWC.
method here given is by means of the compass and a plumb-
line, and is sufficiently accurate for compass-work. The com-
pass-sights are brought into line with the plumb-line and the
pole-star (Polaris), when this is at either eastern or western
elongation. This star appears to revolve in an orbit of i° i8'
radius. Its upper and lower positions are called its upper and
lower culminations, and its extreme east and west positions are
called its eastern and western elongations, respectively. When
it is at elongation it ceases to have a lateral component of
motion, and moves vertically upward at eastern and downward
at western elongation. If the star be observed at elongation,
therefore, the observer's watch may be as much as ten or
fifteen minutes in error, without its making any appreciable
error in the result. The method of making the observation is
as follows :
Suspend a fine plumb-line, such as an ordinary fishing-line,
by a heavy weight swinging freely in a vessel of water. The
line should be suspended from a rigid point some fifteen or
twenty feet from the ground. Care must be taken to see that
the line does not stretch so as to allow the weight to touch the
bottom of the vessel. Just south of this line set two stakes in
the ground in an east and west direction, leaving their tops at
an elevation of four or five feet. Nail to these stakes a board
on which the compass is to rest. The top of this board should
be smooth and level. This compass-support should be as far
south of the plumb-line as possible, to enable the pole-star to
be seen below the line-support. A sort of wooden box may
be provided, in which the compass is rigidly fitted and levelled.
Several hundred feet of nearly level ground should be open to
the northward, for setting the azimuth-stake. Prepare two
stakes, tacks, and lanterns. Find from the table given on page
32 the time of elongation of the star. About twenty minutes
before this time, set the compass upon the board, bringing both
sights in the plane defined by the plumb-line and star. The
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 3I
line must be illuminated. The star will be found to move
slowly east or west, according as it is approaching its eastern
or western elongation. When it ceases to move laterally, the
compass is carefully levelled, the rear compass-sight brought
into the plane of the line and star, and then the forward com-
pass-sight made to coincide with the rear sight and plumb-line.
(If the forward sight were tall enough, we could at once bring
both slits into coincidence with line and star.) Continue to ex-
amine rear sight, line, and star, and rear sight, forward sight,
and line alternately, until all are found to be in perfect coinci-
dence, the instrument still being level. If this is completed
within fifteen minutes of the true local time of elongation, the
observation may be considered good ; and if it is completed
within thirty minutes of the time of elongation, the resulting
error in azimuth will be less than one minute of arc. Having
completed these observations, remove the plumb-line and set a
stake in the line of sight as given by the compass, several hun-
dred feet away. In the top of this stake a tack is to be set
exactly on line. For setting this tack, a board may be used,
having a vertical slit about \ inch wide, covered with white
cloth or paper, behind which a lamp is held. This slit can
then be accurately aligned and the tack set. A small stake
with tack is now set just under the compass (or plumb-line),
and the work is complete for the night. Great care must be
taken not to disturb the compass after its final setting on the line
and star.
At about ten o'clock on the following day, mount the com-
pass over the south stake. From the north stake lay off a line
at right angles to the line joining the two stakes (by compass,
optical square, or otherwise) towards the west if eastern
elongation, or towards the east if western elongation had been
observed. Carefully measure the distance between the two
stakes by some standardized unit. From the table of azimuths
on page 33 find the azimuth of the star at elongation for the
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32
SURVEYING.
given time and latitude. Multiply the tangent of this angle
by the measured distances between the stakes, and care-
fully lay it off from the north tack, setting a stake and tack.
This is now in the meridian with the south point. With the
compass in good adjustment, especially as to the bubbles and
the verticality of the sights, the observation for declination
may now be made. If this be done at about 10.30 A.M., it
will give the mean daily declination. Many readings should
be made, allowing the needle to settle independently each time.
The fractional part of a division on the graduated limb should
be read by the declination-vernier, thus enabling the needle to
be set exactly at a graduation-mark. If all parts of this work
be well done, it will give the declination as accurately as the
flag can be set by means of the open sights.
MEAN LOCAL TIME OF THE ELONGATIONS OF POLARIS.
(This table answers directly for the year 1901 and for latitude 40*.]
Date.
, c
, c
Date.
c 0
S3:
Time.
Date.
5S
Time.
h m ;
k m
Jan. I
W.' 1234.8 A.M. 1
May I
E.
4 48.1 A.M.
" T5
II 39.6 P.M.
" 15
3 52.9 ••
Feb. I
1032.4 "
June I
246.7 **
*• 15
«•
937.2 ••
*• 15
I 51.5 "
Mar. I
841.9 •'
July I
12 49.2 •*
" 15
746.7 '*
'* 15
II 54.0 P.M.
Apr. I
♦639.8 ••
Aug. I
10 47-8 •*
*' 15
E.
*5 44.6 A.M.
*' 15
952.6 •' 1
♦ Probably not visible to the naked eye.
Sept. I
" 15
Oct. I
'• 15
Nov. I
'• 15
Dec. 1
'* 15
ll
Time.
ll m
E.
8 46.4 P.M.
W.
751.2 -
♦648.7 ••
♦5435 A.M.
436.7 ••
«•
341.5 '•
238.6 ••
<<
I 43-4 ••
For the years following igoi, to 1911, add the following
to the times of elongation given in the above table :
190a
4-i".4
1903
1904
J905
1906
1907
1908
+ 5'".7*
+ i'".7
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS.
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34 SURVEYING.
If the elongation of Polaris does not come at a suitable
time for observing for declination, the upper culmination, which
occurs 5** 54".6 after the eastern, or the lower culmination,
6** 03™.4 after the western elongation, may be chosen. The
objection to this is that the star is then moving at its most
rapid rate in azimuth. It is so near the pole, however, that if
the observation can be obtained within two minutes of the
time of its culmination the resulting error will be less than i'
of arc. This will then give the true meridian without having
to make offsets.*
It must be remembered that the time of elongation given
in the table is the local time at the place of observation. In-
asmuch as hourly meridian time is now carried at most points
in this country to the Complete exclusion of local time, it will
be necessary to find the local time from the known meridian or
watch time. Thus, all points in the United States east of Pitts-
burgh use the fifth-hour meridian time (75® w. of Greenwich) ;
from Pittsburgh to Denver, the sixth-hour meridian time (90°
w. of Greenwich), etc. To find local time, therefore, the longi-
tude east or west of the given meridian must be found. This
can be determined with sufficient accuracy from a map. Thus,
if the longitude of the place is 80° w. from Greenwich, it is
5° w. of the fifth-hour meridian, or local time is twenty min-
utes slower than meridian time at that place If meridian time
is used at such a place, the elongation will occur twenty min-
utes later than given by the table. If the longitude from
Washington is given on the map consulted, add it to 'J^^ if
west of Washington, and subtract it from ^^^ if east of Wash-
ington, to get longitude from Greenwich.
USE OF THE NEEDLE-COMPASS.
34. The Use of the Needle-compass is confined almost
* For finding azimuth from Polaris at any hour see Art. 38K/, p. 539. In 1893
Polaris was in the meridian, when it and C Urs« Majoris or Mizar(the middle one of
the three stars in the tail of the Great Bear) come into the same vertical line. For
following years allow a lapse of o™.35 per year, after coming into such vertical line,
for Polaris tocome to the meridian.— /#/^. /, U, S. C, and G. Survey Report, 1891.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS 35
exclusively to land-surveying, where an error of one in three
hundred could be allowed. As the land enhances in value,
however, there is an increasing demand for more accurate
means of determining areas than the compass and chain afford.
The original U. S. land-surveys were all made with the needle, or
with the solar, compass and Gunter's chain. Hence all land
boundaries in this country have their directions given in com-
pass-bearings, and their lengths in chains of sixty-six feet each.
The compass is used, therefore, —
1. To establish a line of a given bearing.
2. To determine the bearing of an established line.
3. To retrace old lines.
If the true bearing is to be used, the declination of the
needle from the meridian must be determined and set off by
the vernier.
If the magnetic bearing is used, the declination of the
needle at the time the survey was made should be recorded
on the plat.
If old lines are to be retraced, the declinations at the times
of both surveys must be known.
The needle should be read to the nearest five minutes.
This requires reading to sixths of the half-degree spaces, but
this can be done with a little practice.
Always lift the needle from the pivot before moving the in-
strument.
If the needle is sluggish in its movements and settles quickly
it has either lost its magnetic force or it has a blunt pivot. In
either case it is likely to settle considerably out of its true posi-
tion. The longer a needle is in settling the more accurate will
be its final position. It can be quickly brought very near its
true position by checking its motion by means of the lifting
screw. In its final settlement, ho\yever, it must be left free.
Careful attention to the instrumental adjustments, to local
disturbances, and close reading of the needle are all essential
to good results with the compass.
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36 SURVEYING.
35. To set off the Declination, we have only to remem-
ber that the declination arc is attached to the line of sight and
that the vernier is attached to the graduated circle. If the
declination is west, then when the line of sight is north the
north end of the needle points to the left of the zero of the
graduated circle. In order that it may read zero, or north, the
circle must be moved towards the left, or opposite to the hands
of a watch. On the other hand, if the declination is east, the
circle to which the vernier is attached should be moved with
the hands of a watch. This at once enables the observer to
set the vernier so that tlie needle readings will be the true
bearings of the line of sight.
36. Local Attractions may disturb the needle by large or
small amounts, and these often come from unknown causes.
The observer should have them constantly in mind, and keep all
iron bodies at a distance from the instrument when the needle
is being read. The glass cover may become electrified from
friction, and attract the needle. This can be discharged by
touching it with a wet finger, or by breathing upon it. Read-
ing-glasses should not have guttapercha frames, as these be-
come highly electrified by wiping the lens, and will attract the
needle. Such glasses should have brass or German-silver
frames. No nickel coverings or ornaments should be near, as
this metal has magnetic properties. A steel band in a hat-
brim, or buttons containing iron, have been known to cause
great disturbance. In cities and towns it is practically impos-
sible to get away from the influence of some local attraction,
such as iron or gas pipes in the ground, iron lamp-posts, fences,
building-fronts, etc. For this reason the needle should never
be used in such places.
In many regions, also, there are large magnetic ij;on-ore de-
posits in the ground, which give special values for the declina-
tion at each consecutive station occupied. It is practically
impossible to use magnetic bearings in such localities.
The test for local attraction in the field-work is to read the
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 37
bearing of every line from both ends of it. If these are not
the same, and no error has been made, there is some local dis-
turbance at one station not found at the other. If there is
known to be mineral deposits in the region it may perhaps be
laid to that. If not, the preceding station should be occupied
again, and the cause of the discrepancy inquired into. If the
forward and reverse bearings of all lines agree except the bear-
ings taken from a single station, then it may be assumed there
is local attraction at that station.
ELIMINATION OF LOCAL ATTRACTIONS.
37. To establish a Line of a Given Bearing, set the com-
pass up at a point on the line, turn ofif the declination on the
declination-arc, and bring the north end of the needle to the
given bearing. The line of sight now coincides with the re-
quired line, and other points can be set.
38. To find the True Bearing of a Line, set the compass
up on the line, turn off the declination by the vernier, bring
the line of sight to coincide with the line with the south part of
the graduated circle towards the observer, and read the north
end of the needle. This gives the forward bearing of the line.
39. To retrace an Old Line, set the compass over one
well-determined point in the line and turn the line of sight
upon another such point. Read the north end of the needle.
If this reading is not the bearing as given for the line, move
the vernier until the north end of the needle comes to the
given bearing, when the sights are on line. The reading of
the declination-arc will now give the declination to be used in
retracing all the other lines of the same survey. If a second
well-determined point cannot be seen from the instrument-sta-
tion, a trial-line will have to be run on an assumed value for
the declination, and then the value of the declination used on
the first survey computed. Thus, if the trial-line, of length /,
comes out a distance x to the right of the known point on
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38 SURVEYING,
the line, the vernier is to be moved in the direction of the hands
X
of a watch an angular amount whose tangent is y. If the
trial-line comes out to the left of the point, move the vernier
in a direction opposite to the hands of a watch.
PRISMATIC COMPASS.
40. The Prismatic Compass is a hand-instrument pro-
vided with a glass prism so adjusted that the needle can be
read while taking the sight. A convenient form is shown in
Fig' 9> which is carried in the pocket as a watch. The line of
Fig. 9.
sight is established by means of the etched line on the glass
cover 5. It is used in preliminary and reconnoissance work, in
clearing out lines, etc.
EXERCISES FOR COMPASS ALONE OR FOR COMPASS AND CHAIN
41. Run out a line of about a mile in length, on somewhat uneven ground,
establishing several stations upon it, using a constant compass-bearing. Then
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ADJUSTMENT, USE, AND CAkE OE INSTkUMENTS. 39
Tin back by the reverse hearings and note how nearly the points coincide with the
former ones. The chain need not be used.
42. Select some half dozen points that enclose an area of about forty acres
(one quarter mile square) on uneven ground. Let one party make a compass-
aad^hain survey of it, obtaining bearing and length of each side. Then let
other parties take these field-notes and, all starting from a common point, run
mttke lines as given by the fieldnotei, setting other stakes at all the remaining
comers, each party leaving special marks on their own stakes. Let each party
plot their own survey and compare errors of closure.
43. Select five points, three of which are free from local attraction, while two
consecutive ones are known to be subject to such disturbance. Make the sur>
Tcy, finding length and forward and reverse bearings of every side. Determine
what the true bearing of each course is, and plot to obtain the error of closure.
44. Let a number of parties observe for the declination of the needle, using
1 common point of support for the plumb-line. Let each party set an inde-
pendent meridian stake in line with the common point. Note the distance of
each stake />v/« the mean position, and compute the corresponding angular dis-
crepancies. (March and September are favorable months for making these
observations, for then Polaris comes to elongation in the early evening.)
The above problems are intended to impress upon the .«tudent the relative
errors to which his work is subject.
THE SOLAR COMPASS.
45. The Burt Solar Compass essentially consists first, of
a polar axis rigidly attached in the same vertical plane with a
terrestrial line of sight, the whole turning about a vertical axis.
When this plane coincides with the meridian plane, the polar
axis is parallel with the axis of the earth. Second, attached
to the polar axis, and revolving about it, is a line of collimation
making an angle with the polar axis equal to the angular dis-
tance of the sun for the given day and hour from the pole.
This latter angle is 90® plus or minus the sun's declination,
according as the sun is south or north of the equator. The
polar axis must therefore make an angle with the horizon
equal to the latitude of the place, and the line of collimation
must deviate from a perpendicular to this axis by an angular
amount equal to, and in the direction of, the sun's declination.
With these angles properly set, and the line of collimation
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40
SURVEYING,
turned upon the sun, the vertical plane through the terrestrial
line of sight, and the polar axis must lie in the meridian, for
Fig.
Otherwise any motion of the line of collimation about its axis
would not bring it upon the sun.
In Fig. lo is shown a cut of this instrument as manufac-
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 4^
tured by Young & Sons, Philadelphia. The polar axis is shown
at/, and the terrestrial line of sight is defined by the slits in
the vertical sights, the same as in the needle-compass. The
line of coUimation is defined by a lens at the upper end of the
arm a^ and a silver plate at the lower end, containing gradua-
tions with which the image of the sun, as formed by the lens,
is made to coincide. The polar axis is given the proper incli-
nation by means of the latitude-arc /, and the line of coUima-
tion is inclined from a perpendicular to this axis by an amount
equal to the sun's declination by means of the declination-arc
d. When these arcs are properly set, the arm a is revolved
about the polar axis, and the whole instrument about its verti-
cal axis, until the image of the sun is properly fixed on the
lines of the silver plate, when the terrestrial line of sight, as
defined by the vertical slits, lies in the true meridian. Any
desired bearing may now be turned off by means of the hori-
zontal circle and vernier, shown at v. The accuracy with
which the meridian is obtained with this instrument depends
on the time of day, and on the accuracy with which the lati-
tude- and declination-angles are set off. It is necessary to at-
tend carefully, therefore, to the
ADJUSTMENTS OF THE SOLAR COMPASS.
46. To make the Plane of the Bubbles perpendicular to
the Vertical Axis. — This is done by reversals about the verti-
cal axis, the same as with the needle-compass.
47. To adjust the Lines of CoUimation. — The declination-
arm a has two lines of coUimation that should be made paral-
lel. As it is shown in the figure, it is set for a south declina-
tion. This is the position it will occupy from Sept. 20 to
March 20. When the sun has a north declination, as from
March 20 to Sept. 20, the declination-arm is revolved 180°
about the polar axis, and a line of coUimation established by
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4^ WrP'eVMG.
a lens and a graduated disk on opposite ends from those pre-
viously used. Each end of this arm, therefore, has both a lens
and a disk, each set of whijch establishes a line of collimation.
The second adjustment consists in making these two lines of col"
limation parallel to each other. They are made parallel to each
other by making both parallel to the faces of the blocks con»
taining the lenses and disks. To effect this, the arm must be
detached and laid upon an auxiliary frame which is attached
in the place of the arm, and which is called an adjuster. With
the latitude- and declination-arc set approximately for the given
time and place, lay the declination-arm upon the adjuster, and
bring the sun's image upon the disk. Now turn the arm care-
fully bottom side up (not end for end) and see if the sun's
image comes between the equatorial lines on the disk.* If not,
adjust the disk for one half the displacement, and reverse again
for a check. When this disk is adjusted, turn the arm end for
end, and adjust the other disk in a similar manner. Having
now made both lines of collimation parallel to the edges of the
blocks, they are parallel to each other.
48. To make the Declination-arc read Zero when the
Line of Collimation is at Right Angles to the Polar Axis.
— Set the vernier on the declination-arc to read zero. By any
means bring the line of collimation upon the sun. When
carefully centred on the disk, revolve the arm 180® quickly
about the polar axis, and see if the image now falls exactly
on the other disk. If not, move the declination-arm by
means of the tangent-screw until the image falls exactly on
the disk. Read the declination-arc, loosen the screws in the
vernier-plate, and move it back over one half its distance
from the zero-reading. Centre the image again, reverse 180®,
and test. This adjustment depends on the parallelism of the
two lines of collimation. If the vernier-scale is not adjustable,
* It would not be expected to fall between the hour-lines on the disk, since
some lime has elapsed.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 43
one half the total movement is the index error of the declina-
tion-arc, and must be taken into account in all settings on this
arc
The two preceding adjustments should be made near the
middle of the day.
49. Toadjustthe Vernier of the Latitude-arc. — Find the
latitude of the place^ either from a good map or by a transit-
observation. Set up the compass a few minutes before noon, ;
with the true declination set off for the given day and hour.
Bring the line of collimation upon the sun, having it clamped
in the plane of the sights, or at the twelve-hour angle, and
follow it by moving the latitude-arc by means of the tangent-
screw, and by turning the instrument on its vertical axis.
When the sun has attained its highest altitude, read the lati-
tude-arc. Compare this with the known latitude. Move the
vernier on this arc until it reads the true latitude ; or, if this
cannot be done, the difference is the index error of the latitude-
arc If, however, the latitude used with the instrument be
that obtained by it, as above described, then no attention need
be paid to this error. This error is only important when the
true latitude is used with the instrument in finding the meridian,
or where the true latitude of the place is to be found by the in-
strument. In using the solar compass, therefore, always use
the latitude as given by that ifistrument by a meridian observa-
tion on the sunJ^
50. To make the Terrestrial Line of Sight and the Polar
Axis lie in the same Vertical Plane. — This should be done by
the maker. The vertical plane that is really brought into the
meridian by a solar observation is that containing the polar
axis, and by as much as the plane of the sights deviates from
* Since the sun may cross the meridian as much as 15 minutes or more
before or after mean noon, this observation may have to be taken that much
before or after 12 o'clock mean time. It is, however, in all cases, an observation
OD the sun ai culmination.
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44 surveyh^g.
this plane, by so much will all bearings be in error. The best
test of this adjustment is to establish a true meridian by the
transit by observations on a circumpolar star ; and then by
making many observations on this line, in both forenoon and
afternoon, one may determine whether or not the horizontal
bearings should have an index-correction applied.
USE OF THE SOLAR COMPASS.
51. The Solar Compass is used on land and other surveys
where the needle-compass is either too inaccurate, or where,
from local attraction, the declination of the needle is too vari-
able to be accurately determined for all points in the survey.
Where there is no local attraction, however, and the declination
of the needle is well known, the advantages of the solar com-
pass in accuracy are fairly offset by several disadvantages in its
use which do not obtain with the needle-compass. Thus, the
solar compass should never be used when the sun is less than
one hour above the horizon, or less than one hour from noon.
Of course it cannot be used in cloudy weather. For such times
as these bearings may be obtained by a needle which is always
attached, but then the instrument becomes a needle-com-
pass simply. It is also much more trouble, and consumes
more time in the field than the needle-compass. But more
'•significant than any of these is the fact that if the adjustments
are not carefully attended to, the error in the bearing of a line
may be much greater by the solar compass than is likely to
be made by the needle-compass, when there is no local attrac-
tion. It is possible, however, to do much better work with
the solar compass than can be done with the needle-com-
pass.
52. To find the Declination of the Sun.— On account of
the inclination of the earth's axis to the plane of its orbit, the
sun is seen north of the celestial equator in summer, and south
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 45
of it in winter. This deviation, north or south of the equator,
is called north or south declination, and is measured from any
point on the earth's surface in degrees of arc.
On about the 2ist of June the sun reaches its most northern
declination, and begins slowly to return. Its most southern
point is reached about December 2ist. In June and Decem-
ber, therefore, the sun is changing its declination most slowly,
while at the intervening quadrant-points of the earth's orbit,
March and September, it is changing its declination most
rapidly, being as much as one minute in arc for one hour in
time. It is evident, therefore, that we must regard the decli-
nation of the sun as a constantly changing quantity, and,
for any given day's work, a table of declinations must be
made out for each hour of the day. The American Ephemeris
and Nautical Almanac gives the declination of the sun for noon
of each day of the year for both Greenwich and Washington.
Since the time universally used in this country is so many
hours from Greenwich, it is best to use the Greenwich declina-
tions. Since, also, we are five, six, seven, or eight hours west
of Greenwich, the declination given in the almanac for Green-
wich noon of any day will correspond to the declination here
2it 7, 6, 5, or 4 o'clock A.M. of the same date, according as East-
ern, Central, Mountain, or Western time is used. As this
standard time is seldom more than 30 minutes different from
local time, and as this could never affect the declination by more
than 30 seconds of arc, it will here be considered sufficient to
correct the Greenwich declination by the change, as found for
the standard time used. Thus, if Central (90th meridian) time
is used, the declination given in the almanac is the declination
at 6 o'clock A.M. at the place of observation. To this must be
added (algebraically) the hourly change in declination, which is
also given in the almanac. A table may thus be prepared, giv-
ing the declination for each hour of the day.
53. To correct the Declination for Refraction.— All rays
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46 SURVEYING.
of light coming to the earth from exterior bodies are refracted
downward, thus causing such bodies to appear higher than
they really are. This refraction is zero for normal (vertical)
lines, and increases towards the horizon. It varies largely,
also, with the special temperature, pressure, and hydrometrical
condition of the atmosphere. Tables of refraction give only
the mean values, and these may differ largely from the values
found for any given time, especially for lines near the horizon.
It is for this reason that all astronomical observations made
near the horizon are very uncertain. There is but one setting
on the solar compass that has reference to the position of the
sun in the heavens, and that is the declination. Now, the re-
fraction changes the apparent altitude of the body ; and by so
much as a change in the altitude changes the declination, by
so much does the apparent declination differ from the true dec-
lination. Evidently it is the apparent declination that should
be set off. When the sun is on the meridian, the change in
altitude has its full effect in changing the declination, but at
other times the change in declination is less than the change
in altitude.
The correction to the declination due to refraction is found
from the following final equations : *
tan N = cot cp cos /,
sin iV ^ ^
tan q = 7-^ X7\ tan /,
^ cos {6 + N) *
cot (d + N)
tan js = ^ \
cos f
dS = — ^fe cos f,
♦ Sec Chauvenet's ** Spherical Astronomy," voL i., p. 171, and Doolittle't
" Practical Astronomy," p. 159.
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A£>JVSTMkNf, Use, ANi> caum op In^THOMMI^TS, A1
where <p = latitude ; / = hour angle from the meridian ; 6 =
declination of sun ; z = zenith distance of sun; iVand g being
auxiliary angles to facilitate the computation.
From these equations we may compute the auxiliary angle
^, and the zenith distance ^, for each hour from noon, for every
day of the year. Then from a table of mean refractions, giving
the refraction for given altitudes, or zenith distances, which is
dz^ we may find the corresponding dd, which is the correction
to be applied to the declination for refraction.
In this manner the following table has been computed for
the latitude of 40°. For any other latitude the correction is
found by multiplying the correction given in the table by the
corresponding coefficient, as given in the table " Latitude Co-
efficients." These coefficients were obtained by plotting the
ratios of the actual refraction at the different latitudes to that
at latitude 40°, for each hour from 7 A.M. to 5 P.M. and for the
various declinations. It was found that this ratio was almost
a constant, except for very low altitudes, where the inherent
uncertainties of an observation are very large, from the actual
refraction varying so largely from the mean, as given in the
tables. A mean value of this ratio was chosen, therefore,
which enables the corrections at other latitudes to be found in
terms of those in latitude 40° without material error. These
ratios are given in the Table of Latitude Coefficients.
EXAMPLE.
Let it be required to prepare a table of declination settings
for a point whose latitude is 38° 30', which lies in the " Central
Time Belt," and for April 5, 1890.
Since the time is 6 h. earlier than that at Greenwich, the
declination given in the Ephemeris for Greenwich mean noon
(6° 9' 57") is the declination for the given place at 6 A.M. If
the point were in the " Eastern Time Belt " it would be the
declination at 7 a.m., etc. Suppose it is desired to prepare
declination settings from 7 A M. to 5 P.M. From the table of -
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48
SURVEYING.
TABLE OF REFRACTION CORRECTIONS TO BE APPLIED TO THE
DECLINATIONS.
Refraction
Refraction
Refraction
Refraction
Date.
CORRECnON.
Date.
Correction.
Datb.«
Correction,
Date.
Correction.
Latitude 4o«».
Latitude 40®.
Latitude 40".
Latitude 4o'».
Jan.
Feb.
Mar.
May.
♦i h. I' 58"
»3
I h. x' 16"
30
1 h. 42"
3 47
3 57
X4
X h. 23"
X
2 2 16
14
2 X 25
31
X5
2 27
a
3 3 04
15
3 I 48
April.
x6
3 34
3
4 6 23
x6
17
4 2 47
5 8 39
X
3
3
\l
i Ai
4
1
1 I 54
2 2 11
3 2 59
x8
19
so
X X 12
3 I 20
3 I 40
4
« 39
a 44
3 54
X9
30
31
X 32
3 36
3 33
\
4 6 01
21
23
4 2 31
5 6 49
\
23
33
4 47
5 X X5
9
I I 51
a3
I T 07
9
t 36
24
1 2X
lO
2 2 07
24
2 I 15
10
3 41
^
2 35
II
3 a 51
as
3 I 33
XX
3 51
3 3a
X2
«3
4 5 40
26
27
4 2 id
5 5 29
12
13
4 I 10
5 X 58
^
4 46
5 X 13
14
I I 46
3 2 01
28
Mar.
I
X I 03
3 X xo
14
• H
39
30
3X
X 3C
3 24
;g
3 2 40
4 5 00
2
3
4
3 » a7
4 3 06
5 4 39
12
4 I 06
5 X 49
June.
X
3
3 3x
4 44
5 III
<9
»9
X 3a
SO
1 I 42
2 X 56
5
I 0 59
20
3 36
3
X X9
21
6
3 X 06
1 21
3 45
4
9 23
23
3 2 31
\
3 I 21
23
4 X 02
5
3 30
23
4 4 35 ,
4 I 56 1
33
5 X 42
6
4 43
9
5 4 04
7
5 X 10
a4
24
I 30
as
I I 37
xo
1 55
'1
2 34
8
I t8
26
2 I 50
XI
3 X 03
36
3 42
9
3 33
37
3 a 22
12
3 I 15
27
4 58
10
3 29
28
4 4 07
13
4 X 47
38
5 X 36
XX
4 4'
M
5 3 34
12
5 X 09
29
29
X 38
30
I
X X 32
a 1 44
3 a 13
4 3 41
16
17
18
' 5a
a 58
3 I 10
4 X 39
iJay.
X
3
3 33
3 39
4 55
5 X 30
X3
X4
X 18
3 33
3 29
4 42
9
»9
5 3 08
3
X7
5 X 08
3
4
X 1 26
30
21
I 48
a S4
4
5
X 36
3 30
x8
X9
X x8
2 22
5
2 I 37
22
3 « 05 1
6
3 37
20
3 29
6
3 2 04
23
4 « 32
2
4 53
21
4 42
7
4 3 ax
24
5 2 5'
5 X 26
32
S X 08
8
I r 21
25
I 45
9
X 25
23
X x8
9
2 X 31
26
a 50
10
2 29
24
2 22
TO
3 I 56 '
a?
3 I o«
IX
3 36
^
3 29
II
^ 3 04 i
28
4 I 25
12
4 51
36
4 42
13
1
1
29
5 2 34
»3
5 I 32
27
5 I 08
♦ The hours are counted each way from noon,
the 3d hour in the toblc.
Thus 9 A.M. ai)d 3 p.m. would correspond to
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SURVEYING.
48a
Refraction
Refraction
Refraction
Refraction
Date.
Correction.
Date.
Correction.
Date.
Correction.
Datk.
Correction.
Latitude 40*.
Latitude 40*'.
Latitude 40°.
Latitude 40*>.
June.
Aug.
Oct.
Nov.
38
29
X h. x8"
2 22
;?
1 h. 32"
2 36
6
I
9
1 h. 1' 03"
2 I 10
20
21
1 h. i' 46'
2 2 ox
jSy
3 29
4 43
5 i'o9
«9
20
3 45
4 i' 02
4 a oA
22
23
3 2 40
4 4 59
X
21
5 » 4a
xo
5 4 39
24
3
«9
aa
23
' ^.
II
I I 07
S
4
2 23 i
94
' *!
12
a I 15
I I 50
s
3 30
1 25
4 I 06
»3
3 I 33
4 2 x8
27
28
2 2 06
6
4 43
a6
5 I 49
14
3 2 49
7
5 X xo
37
36
15
5 5 29
29
4 5 33
8
X ao
28
2 41
9
2 24
29
3 51
16
X 1 12
10
.3 31
30
4 I XO'
*Z
2 I 20
^.
XI
4 44
3«
5 X 58
18
3 I 40
> I 54
12
5 X II
19
4 2 31
X
2 2 XI
Sept.
20
5 6 49
2
3 2 59
«3
X 21
X
I 39
3
4 6 01
«4
2 25
2
2 44
4
;i
3 32
3
3 54
21
I X 16
4 46
4
5 3 c8
22
3 ;:i
'7
5 « 13
5
23
5
6
24
4 2 47
I 1 58
18
I 22
6
I 42
25
5 8 39
\
2 2 16
»9
2 26
?
a 47 •
3 3 04
ao
3 33
3 57
9
4 6 23
2l
4 47
9
4 X 10
5 9 18
26
I X ai
»
5 I '5
10
%
a I -^i
3 I 56
»3
X 23
11
X 48
29
4 3 04
10
I 2 00
«4
2 27
12
2 50 1
30
5 11 01
11
2 2 19
as
3 34
«3
3 I 01
12
26
27
J rt
14
'5
4 1 25
5 2 34
iJov.
X I a6
2 I 37
13
28
16
I 48
T
2
3 2 04
29
30
' 25
3 3I
4 51
5 X 22
X 26
;i
2 54
3 I 05
3
4
4 3 21
5 13 57
;i
1 2 OT
2 ^20
X
3
»9
20
21
32
4 » 33
5 2 51
' 52
58
7
1 I 3a
2 X 44
3 2 13
4 3 41
19
3 3 11
4 6 47
3
a 30
23
3 I 10
8
20
I 2 01
4
3 37
24
\ 3^
1 55
2 I 02
3 1 »5
' 9
21
2 2 20
1
1
X 38
2 32
85
26
xo
II
12
« I 37
2 I 50
3 2 22
4 4 07
22
23
24
3 3 "
4 6 49
9
3 39
29
4 I 47
13
25
I 2 00
xo
4 55
30
5 3 34
»4
26
a a 19
3 3 09
4 6 13
XI
5 I 30
Oct.
'1
12
> 30
X
X 59
15
1 I 42
2 I <6
29
'3
a 34
3
a X 06
16
M
3 43
3
3 I 21
17
3 2 31
4 4 35
15
4 58
4
4 I 56
18
30
16
5 I 36
5
5 4 04
«9
3
Difffl?ed byCjOOgtC
4^^ ADJUSTMENT, UsE, AND CAkE OF INSTkVMENTS.
TABLE OF
LATITUDE COEFFICIENTS.
Latitude.
Coefficient.
Latitude.
Coefficient.
Latitude.
Coefficient
15°
.30
30"
.65 1
45'
1.20
i6
.32
31
.68
46
1.24
17
•34
32
.71
47
1.29
18
.36
33
•75 ,
48
1.33
19
38
34
.78 I
49
1.38
20
40
35
.82 1
50
1.42
21
42
36
•?5
51
1.47
22
44
37
.89
52
1.53
23
46
38
•92 .
53
1.58
24
48
39
.96
54
1.64
as
50
40
1. 00
55
1.70
26
53
41
1.04
56
1.76
27
56
42
1.08
57
1.82
28
59
43
1.12
58
1.88
29
.62
44 I. 16
59
1.94
Note. — For any other latitude than 40° the refraction corrections given in the
preceding table are to be multiplied by the coefficients given in this table to obtain
the true refraction corrections for that latitude.
latitude corrections we find that the refraction corrections will
be .94 of those given in the table. The following table of
declination settings may now be made out :
Hour.
Declination.
Refr. Cor.
Setting.
Hour.
Declination.
Refr. Cor.
Setting.
7
+ e** 10' 54"
+ a' 00"
+ (P 12' 54"
I
+ (P 16' 35"
+ 37"
+ 60 1/ la"
8
6 II s«
+ X xo
6 13 01
a
6 17 31
+ 4X
6 x8 12
9
6 IS 47
+ 5«
6 13 38
3
6 18 a8
+ 5«
6 19 19
10
^ '3 44
+ 41
6 14 as
4
6 19 as
+ \' 10"
6 ao 3S
II
6 14 41
+ 37
6 15 18
5
6 ao aa
+ a 00
6 aa aa
From March 20th to September 20th the declination is
positive, while from September 20th to March 20th it is nega-
tive. From December 20th to June 20th the hourly correction
is positive, while from June 20th to December 20th it is nega-
tive. The refraction correction is always positive. Particular
attention must be given to all these signs in making out the
table of declination settings.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 48^
53a* A Simple Way of Correcting the Sun's Declination
for the Effect of Refraction.* — In case the observer should
not have with him the refraction tables, found on the preced-
ing three pages, the refraction can be determined by the aid of
his watch, as follows:
Having focussed the eyepiece and object-glass of the tran-
sit so that a clear image of both the sun's disc and the cross-
wires can be seen on the screen held behind the eyepiece (a
piece of white paper held by the hand will do for a screen), set
the horizontal circle of the transit to read some integral ten
minutes and point on the sun by the lower motion.
The earth's diurnal motion will carry the sun across the
vertical thread of the instrument. Note the time on a watch
to the nearest second when the sun is tangent to the vertical
wire. Keeping the lower motion clamped, unclamp the upper
and turn the alidade in the direction of the sun's movement,
i.e., toward the west, and set the vernier to read the next ten
minutes. Note again the time when the sun is tangent to the
vertical thread. Also read the vertical angle to the sun. Then
if we call n the interval of time elapsed in seconds while the
sun (really the earth) was passing through ten minutes of
arc, and call h the vertical angle in degrees, the refraction d^ in
minutes, is given by the equation
2000
h , n
Experience in using this formula has shown that its maxi-
mum errors will not exceed 15" when the sun is above 10^
altitude, while its average error is less than half this amount.
As the refraction correction, as ordinarily computed, is based
upon average conditions of temperature and barometric pres-
sure, seldom exactly realized in any given case, the writer has
* For a very complete explanation of this method see ** Studies in
Astronomy," by Prof. George C. Comstock, University of Wisconsin.
Bulletin No. 3. Vol. I of Science Series.
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4Sd
SURVEYING.
not been surprised to find that results obtained by the use of
the above formula are quite as good as those obtained from
the more complicated and pretentious formulas and tables.
A still more accurate determination of the refraction can be
made by the use of the following equation :
n
where d and n stand for the same quantities as before, and N is
obtained from the following table by entering it with the meas-
ured altitude of the sun as an argument.
k
N
Dif.
for jO.
k
N
Dif.
for «•.
id"
218
15
30-
60
2.3
15'
143
8
40-
37
1.5
20*
103
5
50-
22
I.O
25'
78
3.6
6o*
12
0.7
30-
60
7o-
5
The altitude of the sun need only be observed to the nearest
half degree.
The tabulated values of N correspond to a temperature of
50° F. and a barometric pressure of thirty inches. They may
be adapted to any other temperature by diminishing d by one
per cent for each five degrees by which the temperature ex-
ceeds 50°, or by increasing one per cent for each five degrees
below 50°.
This correction and the correction for variations of the
barometer can usually be neglected. At great elevations, how-
ever, the barometric pressure becomes so much reduced that
its variation must be taken into account, and this is done by
diminishing d by one per cent for each three hundred feet of
elevation above the sea.
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ADJUSTMENT, USE, -AND CAkE OF INSTRUMENTS. 49
54. Errors in Azimuth due to Errors in the Declina-
tion and Latitude Angles. — The spherical triangle involved
in an observation by the solar compass is shown
in Fig. 1 1, where P is the pole, Z the zenith, and cc:
5 the sun. Then
the angle at P=t, the hour-angle from the
meridian ;
" " Z=^ A, the azimuth from the north ^
point ;
" " 5 = q, the variable or parallactic
angle.
Also, the arc PZ = the co-latitude = 90° — 0 ;
*' PS = the co-declination = 90° — d ;
*•' ZS = the co-altitude, or zenith dis- (^)^p
tance = 90^ - A. ^'^- "'
Taking the parenthetical notation of the figure, we have,
from spherical trigonometry,
cos (a) = cos (c) cos (^) + sin (c) sin {d) cos {A).
But in terms of <y, 0, A, and Ay this becomes
sin <y = sin 0 sin A + cos 0 cos A cos A.
U)
In a similar manner, from two other fundamental equations
of the spherical triangle, we may write
cos S cos / = cos 0 sin A — sin 0 cos A cos A ; (2)
cos <y sin / = cos A sin ^. (3)
If we dififerentiate equation (i) with reference to A and d.
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50 su/^vicvixo.
and then with reference to A and 0, we obtain, after some
reductions by the aid of equations (2) and (3)
*dA,= ±t-, (4)
cos 0 sm / ^^'
and dA^= + ^ , (5I
* cos 0 tan / ^^^
Now, if the change (or error) in 6 and 0 be taken as i minute
of arc, or, in other words, if the settings for declination or lati-
tude be erroneous by that amount, either from errors in the
instrumental adjustments or otherwise, then equations (4) and
(S) show what is the error due to this cause in the azimuth, or
in the direction of the meridian, as found. In the following
table, values of dAi and dA^ are given for various values of
0 and / (latitude and hour-angle). In this table no attention
is paid to signs, as it is intended mainly to show the size of the
errors to which the work is liable from inaccurate settings for
declination and latitude ; the values may, however, be used as
corrections to the observed azimuths from such inaccuracies by
observing the instructions in the appended note.
* dAh signifies the change in A due to a small change. dS, in 6, the other
functions involved in equation (i) remaining constant. Similarly for dA ,
when <p alone changes. The derivation of equations (4) and (5) involves a
knowledge of the infinitesimal calculus.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 5 1
TABLE OF ERRORS IN AZIMUTH (BY SOLAR COMPASS) FOR i'
ERROR IN DECLINATION OR LATITUDE.
HOUK.
For 1' Error in Declination.
For x' Error in Latitudr.
Lat. 3o«
Lat. 40»
Lat. 5o«
t
1 Lat. 3o«
Lat. 4o«
Lat, 5o»
11.30 AM. )
12.30 PM. )••
8'.85
lO'.O
• I2'.9
8'.77
9'92
ll'.8
II A.M. )
I P.M. S
446
5 05
6.01
4.33
4.87
5.80
10 A.M. 1
2 P.M. f
2.31
2.61
3."
2.00
2.26
2.70
9 A.M. )
3 PM. f • •
1.63
1.85
2 .20
1. 15
1.30
1.56
8 AM. )
4 PM. )
1.34
1. 51
1.80
0.67
0.75
0.90
7 A.M. )
5 PM. i^
1.20
1.35
1. 61
0.31
0.35
0.37
6 A.M. )
6 P.M. S
1.15
1.30
1.56
0.00
0.00
0.00
Note. — Azimuths observed with erroneous declination or co-latitude may be
corrected by this table by observing that for the line of collimation set too high,
the azimuth of any line /n?/w the south point in the direction S.W.N. E. is
found too small in the forettoon and too large in the afternoon by the tabular
amounts for each minute of error in the altitude of the solar line of sight. The
reverse is true for this line set too low.
Several important conclusions may be drawn from this table
and from equations (4) and (5).
First — That the solar compass should never be used between
II A.M. and I P.M., and preferably not between 10 A.M. and 2
P.M., if the best results are desired.
Second — That at 6 o'clock a.m. and P.M., when the line of col-
limation lies in a plane at right angles to the plane of the me-
ridian, no small change in the latitude-arc will affect the accu-
racy of the result.
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52 SURVEYING.
Third — From equations (4) and (5), we see that the errors
from declination and from latitude have opposite signs, or
that errors from like erroneous settings in declination and
co-latitude have the same sign. Therefore, if the declination-
angle be erroneously set off, and the co-latitude-angle be also
affected by an equal error in the opposite direction, then the two
resulting errors in azimuth will tend to compensate. From the
table it may be seen that for the same latitude and hour-angle
they would nearly balance each other numerically. If, there-
fore, the declination-angle be affected by an error, and the co-
latitude of the place then found by a meridian observation with
the compass, the error of the declination will appear in the re-
suiting co-latitude, with the opposite sign. In this way any con-
stant error in the declination-angle may be nearly eliminated.
Fourth — The best times of day for using the solar compass are
from 7 to 10 A.M. and from 2 to S P.M. So far as the instru-
mental errors are concerned, the greater the hour-angle the
better the observation ; but when the sun is near the horizon,
the uncertainties in the refraction may cause unknown errors
of considerable size.
Fifth — That for a given error in the setting for declination or
latitude the resulting error in azimuth will have opposite signs
in forenoon and afternoon. For, in equations (4) and (5), the
hour-angle, /, has different signs before and after noon ; and
therefore sin t and tan t change sign, thus changing the sign
of the expression. If, therefore, a io-o*clock azimuth is in
error 5' in one direction from erroneous settings, a 2-o*clock
observation with the same instrument should give an azimuth
5' in error in the opposite direction.
55. Solar Attachments are appliances fitted upon transit-
instruments for the purpose of finding the meridian, the same
as is done by the solar compass. The principles of construc-
tion and use are the same as those of the solar compass, the
application of these principles being quite various, however,
giving rise to several forms of attachments, some of which will
be discussed in connection with the transit. Their adjust-
ments and limitations are nearly the same as those here given.
ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, S3
EXERCISES WITH THE SOLAR COxMPASS.
56. Determine a true meridian line by an observation on a circumpolar
star or otherwise, by either the compass or transit. Set the solar compass up
on one point of this line with a target set at another point on the established
meridian. Having carefully adjusted the compass, set the declination-arc to the
right angle for the given day and hour, corrected for refraction, and make a
meridian observation on the sun for latitude. If the true latitude of the place is
known, the diflference will be the index error of the latitude-arc. Leave the lati-
tude-arc set at the readings obtained by the meridian observation (whether it is
the true latitude or not), and make a series of determinations of the meridian by
the compass at various times of day. These will usually be in error from the
true meridian by small amounts. Determine the size of these errors by turn-
ing upon the target and reading the horizontal circle. Record these errors,
with the time of day and name of observer. Each student should make a
series of such observations, determining for himself the errors to which the
work is liable. The same meridian may be used for all, after it has been prop-
erly checked by duplicate observations.
57. Set the latitude- or declination-angle say 3' from its true value, and
observe at various hours of the day, and see if the resulting errors in azimuth
are about three times those given in the table. Note that these resulting errors
are in opposite directions and equal in amount in fore- and after-noon observa-
tions.
58. With the solar compass on the meridian as before, select a series of
points, six or more, whteh are fixed and plainly visible through the slits. Find
the bearing to each of these points by a separate observation on the sun in
each case, paying no attention to the target on the true meridian. Remove the
solar compass and let another student, ignorant of the first bearings, set the
ordinary needle compass over the same point. Bring the line of sight upon
the target and make the needle read south by moving the vernier on the decli-
nation-arc. In other words, set off the declination of the needle. The bear-
ings given by the needle compass should now agree with those obtained by the
solar compass. Read upon the series of selected points, obtaining the bear-
ings to the nearest five minutes. Let a third student take a transit (or the solar
compass would do) and find the true bearings of the selected points with refer-
ence to the established meridian. Compare results and so obtain some data
for determining the relative accuracy of the solar and the needle-compass.
The mean of two azimuths by the solar compass taken on the same line at
equal intervals from noon should be the true azimuth of the line if the instru-
ment has not changed its adjustments in the mean time. This is the way to
&nd the true azimuth of a line by the solar compass.
59. Run a line over a series of points (six or more) in the forenoon by the
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54 SURVEYING,
solar compass, and determine the bearings. In the afternoon run it backagrain,
using the bearings obtained in the forenoon^ set other stakes where the points are
not coincident with the old ones, and note the residual discrepancy at the close
of the work. Divide this by twice the length of the line, and this is the error
of closure due to erroneous bearings. The chain may be used on the first run-
ning of the line, but on the retracing the stakes may be set opposite the first
ones, if not coincident. The object is to determine how much o! the error of
closure in surveying may be attributed to erroneous bearings.
Do the same with the needle compass and compare results. The points
need not be in line, nor need they enclose an area.
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ADJUSTMENT, USE, AKD CA^n OF INSTRUMENTS, 55
CHAPTER III.
INSTRUMENTS FOR DETERMINING HORIZONTAL LINES.
PLUMB-LINE AND BUBBLE.
60. The Plumb-line and the Bubble-tube are at once the
most simple, universal, and essential of all appliances used in
surveying and astronomical work. Without them neither the
zenith nor the horizon could be effectually determined, and the
determination of altitudes and of horizontal lines and planes
would be out of the question. Even azimuths, bearings, and
horizontal angles require that the circle by which they are ob-
tained shall be brought into a horizontal position.
The direction of the plumb-line is by definition a vertical
line, pointing to the zenith, and a plane at right angles to this
line is for that point a horizontal plane. As no two plumb-
lines can be parallel, so no two planes, respectively horizontal
at two different positions on the earth's surface, can be par-
allel.
Parallel horizontal planes can only be planes at different
elevations, all horizontal for a single position on the earth's
surface.
A level surf ace is a surface (not a plane) which is at every
point perpendicular to a plumb-line at that point. If the
earth were covered with a fluid in a quiescent state, the sur-
face of this fluid would be a level surface. This surface would
not be a true oblate spheroid, but would in places vary several
hundred feet from such a mean spheroidal surface. This is
owing to the fact that the earth is not a homogeneous body.
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56 SURVEYING.
thus causing the centre of mass to deviate from the centre of
form. Owing also to much irregularity in the distribution of
the mass, with respect to the form of the earth, there are many,
irregular deviations of the plumb-line* from any one point. A
level surface follows all such deviations.
A bubble-tube is a round glass tube bent or ground so that
its inside upper surface is circular on a longitudinal section.
This is nearly filled with ether, the remaining space being
occupied with ether-vapor, which forms the bubble. This
tube is usually graduated to assist in determining the exact
position of the bubble in the tube. If the tube has been
ground to a perfect circular longitudinal section, then a longi-
tudinal line tangent to this inner surface at the entire of the air-
bubble is a level line, in whatever part of the tube the bubble
may lie. If this were not a level line, the centre of gravity of
the bubble would not occupy its highest possible position and
would move until it did. Since the position of the centre of
a bubble in a tube is determined by reading the position of its
ends and taking the mean, it is necessary that the arc shall be
of uniform curvature — that is, circular.
A line tangent to the inner surface of the bubble-tube at
its centre, as defined by the graduations (or another line parallel
to it) is called ///^ axis of the bubble. When the bubble is in
the centre of the tube, therefore, its axis is horizontal.
Proposition I, If a bubble-tube be rigidly attached to a
frame, and if this frame be reversed on two supports lying in
the vertical plane through the bubble-axis, the supporting
points are level when the bubble occupies the same portion of
the tube in both positions of frame, whether this be the centre
* In the northern portion of the United States, in the vicinity of the Great
Lakes, deviations of the plumb-line (Clarke's Spheroid being used) have been
tound as great as lo or 12 seconds of arc. See Primary Triangulaiion of the
U. S. Lake Survey.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 5/
or not ; providing, of course, that the points of support on the
frame were identical in the two cases.
For, the tangent horizontal lines being identical in the two
positions of the bubble, the vertical distances from this line to
the points of support must be equal, otherwise the direct and
reversed positions would not give identical tangent lines. The
points of support are therefore in a horizontal line.
Proposition I L If a bubble-tube be revolved about an axis
in such a way that the bubble keeps a constant position in the
tube, the axis of revolution is vertical.
For, since the bubble-tube maintains a constant inclination
to the horizon (this inclination being zero when the bubble is
in the centre), the plane of motion can have no vertical com-
ponent, and, therefore, the axis of revolution must be vertical.
Cor. I. Similarly we may say that if a bubble-tube be re-
volved 1 80° about an axis, and if the bubble have the same
reading in the two positions, then the plane of revolution has
no vertical component in the direction of the bubble-axis, and
therefore the axis of revolution lies in a vertical plane at right
angles to the bubble-axis. If the same test be made for any
other two horizontal positions 180° apart (preferably 90° from
first position) and the bubble have the same reading in the
two cases, then the axis of revolution lies in a vertical plane at
right angles to these new positions of bubble-axis, and there-
fore it lies at the intersection of these two vertical planes, or it
is vertical. If two bubble-tubes (not parallel to each other
and preferably at right angles) be rigidly attached to a frame
that revolves about an axis, and if each bubble has the same
reading in two positions of frame 180° apart, the axis of revo-
lution is vertical, even though the two bubbles do not read
alike nor either is at the middle of its tube.
Cor. 2. In all cases where a bubble-tube has been shifted
180® in the same supports, or axis, the angular difference
between the two positions of the bubble is twice the angulal
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58 SURVEYING.
deviation of the supports from a horizontal, or of the axis from
a vertical.
6i. The Accurate Measurement of Small Vertical An-
gles is accomplished by means of the bubble with greater read-
iness and precision than by any other device known. For this
purpose the bubble-tube should be ground accurately to the arc
of a circle with a long radius, and uniformly graduated. Then
a given bubble-movement in any part of the tube corresponds
to a known angular change, when the angular value of a move-
ment of one division in the graduated scale has been deter-
mined. These graduations are usually made on the top of the
glass tube. To measure a small angle by means of the bubble,
read the two ends of the bubble to divisions and tenths, and
take the one half-diflference of end readings.* Shift the bubble
a given amount and read both ends again, taking one half the
difference. The difference of these two results in divisions of
the scale, multiplied by the angular value of one division on
the scale, is the vertical angle through which the tube was
shifted.
62. The Angular Value of One Division of the Bubble
may be found in various ways.
{a) By a telescopic line of sight. Attach the bubble-tube rig-
idly to a mounted telescope, putting the bubble-axis in the plane
of the telescope. Measure off a convenient base-line on level
ground of from 200 to 500 feet. Set the telescope at one end
of this base, and hold a rod vertically at the other. Bring the
bubble near one end of its tube by moving the telescope verti-
cally, and read the two ends. Read the height of the cross-
wires on the rod. Bring the bubble near the other end of tube
and read both the bubble and rod. Repeat many times. Re-
duce the work by taking the half-difference of the two end
* Bubbles are read from the middle outwards towards the ends. Then the
half-difference of end readings is the distance of the centre of the bubble from
die centre of the scale.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 59
readings in each case, thus giving the distance of the centre of
the bubble from the centre of tube for each position. Take
the mean of these results for each set of end readings sepa^
rately. If these mean results were for opposite ends of the
tube, add them together and this gives the average movement
of bubble. Similarly take the mean of the upper readings and
the mean of the lower readings on the rod, and take the differ-
ence, and this is the average movement of the line of sight.
Calling the bubble-movement in divisions of scale Z>, the move-
ment on the rod, in feet, i?, and the length of the base, in feet,
B, we would have, in seconds of arc,
angular value of i div. of bubble = r>n - — 77- *
^ £D sm r
(b) By a large vertical circle. Mount the bubble rigidly
upon the circle, having its axis parallel to the plane of the
circle. Move the bubble from end to end of tube, as before,
reading the corresponding angular changes directly upon the
circle. Divide the mean angular movement by the mean
movement of bubble.
This requires a large circle with micrometer attachments,
such as is used on astronomical instruments.
{c) By a level trier. This consists of a beam hinged at one
end and moved vertically by means of a micrometer screw at
the other. The bubble-tube is placed upon the beam, and the
bubble moved back and forth by means of the screw, each
revolution of which gives a known angular movement to the
beam.
63. General Considerations. — A bubble is sensitive direct-
ly as the length of the radius of curvature, or indirectly as its
rate of curvature. It is also sensitive in proportion to its
length, a long bubblef settling much more quickly and ac-
* Log sin i" = 4.6855749.
f This refers to the length of the air-bubble itself, and not to the glass tube.
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6o SURVEYING.
curately than a short one. Some bubble-tubes have a cham-
ber at one end connected with the main space by a small hole
through the bottom of the dividing partition. This enables
the length of the bubble to be under control. As ether ex-
pands and contracts very largely with temperature, the bubble
is apt to be too long in winter and too short in summer if the
chamber is not used. The bubble-tube should not be rigidly
confined by metallic fastenings about its centre, if the value of
one division is significant, as the changes of temperature will
change its curvature. Bubble-tubes, or level-vials as they are
often called, may be sealed by glass stoppers set in a ghie
made by dissolving isinglass in hot water, and covering with
gold-beater's skin set with the same glue, the whole varnished
over when dry.
THE engineer's LEVEL.
64. The Engineer's Level consists of a telescopic horizon-
tal line of sight joined to a spii it-level, the whole properly
supported and revolving on a vertical axis. Such an instru-
ment is shown in Fig. 12. The vertical parts of the frame
which support the telescope are called wyes, and the cylindri-
cal bearings on the telescope-tube are called the pivot-rings.
The telescope can be lifted out of the wyes by loosening the
clips over the rings, tliese being held by the small pins
attached to strings and shown in the cut. A clamp and
tangent-screw are connected with the axis for holding it to a
given pointing or for moving it horizontally while clamped.
The attached bubble enables the line of sight in the telescope
:o be brought into a horizontal position.
The construction of the instrument is best shown by the
sectional view given in Fig. 13.
The objective is a compound lens, the two parts having
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 6 1
diflferent refractive powers in order that the image may be flat,
A simple lens gives a spherical image. The image is formed
at the plane of the cross-wires, which are attached to the reti.
cule held in place by the capstan-screws shown in the cut. The
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SURVEYING.
line of sight is the line joining the two corresponding points
in object and image with which the intersection of the cross
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ADJUSTMEiWT, USE, AND CAKE OF INSTRUMENTS. 63
wires coincides.* Evidently this line of sight may lie any-
where in the field of view within the Umits of movement of
the reticule. The Itne of collimation is simply the true posi-
tion of the line of sight. The eye-piece serves only to mag-
nify the image, and sometimes to invert it, as is the case in
the sectional view of Fig. 13. The image itself is always
inverted ; and if this be examined by an eye-piece of two
lenses, which simply magnifies but does not invert, the ob-
ject is seen in an inverted position. If four lenses are used
in the eye-piece, it re-inverts the image so that the object is
seen erect. This results in a loss of light and of distinctness.
ADJUSTMENTS OF THE LEVEL.
65. To make the Line of Sight parallel to the Axis of
the Bubble.
First, or Indirect , Method, — This method rests on the
proposition that if two Hnes are parallel to a third line, they are
parallel to each other. This method is indirect, but the
manipulations are readily performed. It is the usual method,
and is frequently given as two separate adjustments.
First, bring the line of sight to coincide with the centres of
the pivot-rings by revolving the telescope, bottom side up, in
the wyes, ar^d' adjusting the reticule until the intersection of
the wires remains on a fixed point of the image.f If the
*Morc correctly, it is the line joining the inner piincipal point of the objec-
tive with that point of the image covered by the intersection of the cross-w^res.
Sec Fig. 61, and note to same.
f The optical axis of a lens is the line joining the centres of the true spherical
surfaces bounding it. If this axis is not coincident with the axis of the tele-
scope, or rings, owing to an erroneous adjustment of ilie objective slide by the
screws near the centre of the telescope tube, Fig. 13, or the improper setting of
the lens in its case, then the image will be shifted laterally a small amount equal
to the lateral deviation of the two ** principal points" of the lens from each other.
In this case the image itself will appear to rotate as the telescope is revolved.
If now the centre of the cross-wires be moved so as to remain on a fixed portion
of the image, it no longer occupies the axis of the telescope, but the line of sight
is uoiK paralUI to this axis, so that this adjustment still accomplishes all that is
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64 SURVEYING,
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instrument gives an erect view of the object, there is one '
inversion between the wires and tlie eye, and therefore the
reticule must be moved in the direction of and one half the
amount of its apparent displacement. If the view is inverted,
there is no inversion between wires and eye, and therefore its
apparent is its true displacement.
Second, make the axis of the bubble-tube parallel to the
bottoms of the rings by reversing the telescope end for end
in the wyes and adjusting the bubble until it remains in the
centre of the tube for the two positions. The telescope
should be removed and replaced with great care so as not to
disturb the relative elevation of the wyes by any jar or shock.
The axis, of course, should be clamped to prevent any hori-
zontal motion in making either part of this adjustment.
This method is based on an assumption which may or may
not be true. It is that the pivot-rings are of the same size,
and therefore the lines joining their centres and bottoms are
parallel.
To find the relative size of the pivot-rings, use a stridtng-
level resting on the two pivot-rings and read in reversed posi-
.ions. Then change the rings in their supports and read the
;evel again in reversed positions. To reduce the notes, the
value of one division of the striding-level must be known.*
The objective is always properly centred and adjusted when
the instrument leaves the maker's hands; but it is apt to
become loose in its frame, and this frame also loosens in the
telescope-tube. If the glass is loose in its frame, unscrew it
from the telescope-tube and screw up the tightening band
desired. Or, the objective may have its optical axis coincident with that of the
telescope and the optical axis of the eye-piece not parallel to that of the objec-
tive, and this will cause the image and wires to appear to rotate together— when
the telescope is revolved. This need cause no error in the work, but should be
adjusted by the screws shown just back of the capstan screws, Fig. 13.
♦ See adjustments in Precise Levelling. Chap. XIV.
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ADJUSTMENT, USE, AND CARE OF INSTJWMENTS. 65
from the rear side. Do not take the glasses apart under any
circumstances, for they are ground for a given relative position
and would not be true for any other. A loose objective is a
fatal defect in a levelling-instrument and must be constantly
guarded against.
Second, or Direct, Method, — This consists in adjusting the
bubble directly to the line of sight, whether this be in the cen-
tre of the pivot-rings or not. It is sometimes called the
" peg adjustment." Drive two pegs on nearly level ground
about 200 feet apart. Set the level about eight or ten inches
from one of them, or so that when the rod is held upon it in a
vertical position the eye end of the telescope will swing about a
half inch from its face. Turn the eye end of the telescope upon
the graduated face of the rod, the bubble being in the middle of
its tube; look through the object end and set a pencil-point on
the rod at the centre of the small field of view, which should
be from i to J inch in diameter. Read the elevation of this
point, which we will call a. Hold the same rod on the distant
peg and, with the bubble in the middle, set the target on the
line of sight, and call this reading b. Now carry the instru-
ment to the distant peg, set it near it, read the elevation of the
instrument as before, which reading we will call a' ; carry the
rod to the first peg and set the target on the line of sight, giv-
ing the reading V. If the line of sight had been parallel to the
axis of the bubble in each case, it would have been horizontal
when the bubble was in the middle of the tube, and hence the
difference between the a and b readings in each case would
have been the difference of elevation of the pegs.* We
should therefore have had
a^b=^b'-a' (i)
*This assumption neglects the effect of the earth's curvature. This is
eight inches to one mile, and is proportional to the square of the distance. For
200 feet it would be about o.ooi of a foot, and twice this, or 0.002 of a foot, is
the error made in the above assumption.
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66 SURVEYING.
If the line of sight was not parallel to the axis of the bub-
ble, however, then the differences of elevation of the two pegs,
as obtained by the two sets of observations, are not equal, and
we should have
(a^b)--(P' ^a')^d (2)
Now d is twice the deviation of the line of sight from the
bubble-axis for the given distance. (Let the student construct
a figure and show this.) If, therefore, the target be moved
up or down as the case may be, a distance equal to \dy then
the line of sight may be brought to this position by the
levelling-screws, and the bubble adjusted to bring it to the
middle, or else the instrument may be left undisturbed with
the bubble in the middle, and the line of sight adjusted to
read upon the target by moving the reticule. The significant
fact is that by moving the target \d from its last position a
true horizontal line is established, and either the bubble or the
line of sight can be adjusted to it after the other has been
brought into a horizontal position by means of the levelling-
screws. Equation (2) may be written
{a^a')-{b^V)=d; (3)
from which it may be seen at once that the line of sight
inclines down when d is positive, and up when d is negative.
We may therefore have for setting the target the following
Rule: Add together the two heights of instrument and the
two rod readings, subtract the latter from the former y and take
one half the remainder. Move the target by this amount from
the b' reading, up when positive and down when negative. It is
then in a horizontal line with the cross-wires of the instrument.
It will be noted that no distances are measured in the above
method as is usually prescribed in peg-adjustments. After
adjusting either the line of sight or the bubble at the second
peg, return to the first peg, read height of instrument again,
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 67
and then read the rod on the second peg for a check. See if
this new value of (a — b) agrees with the adjusted value of
(p' — a^). If not, adjust again.
This method is independent of the relative size of the pivot-
rings and of the condition of the objective. (The objective
must liave a fixed condition or no adjustment is worth any-
thing.) Although the essential relation of parallelism is ob-
tained between the line of sight and the bubble, it must not
be expected that the telescope can be reversed in the wyes or
revolved 180° about its axis without both these auxiliary
adjustments appearing to be in error. For inasmuch as these
two lines have been made parallel without reference to the
axis of the telescope or to the bottoms of the rings, they
probably are not parallel to either of these. If the first meth-
od is used and the adjustment made, it should stand the test
of the second (the necessary assumptions being true), but if
adjusted by the second method it should not be expected to
stand the test of the first method. At the same time the
second method is absolute, while the first is based on assump-
tions that are often untrue. This adjustment should be exam-
ined every day in actual practice.
66. To bring the Bubble-axis into the Vertical Plane
through the Axis of the Telescope. — Turn the telescope
sHghtly back and forth in the wyes, and note the action of the
bubble. If it remains in the centre the adjustment is correct.
If not, move one end of the bubble by means of the lateral
adjusting-screws. If this adjustment is very much in error it
should be made approximately right before going on with the
preceding adjustment.
67. To make the Axis of the Wyes perpendicular to
the Vertical Axis of the Instrument. — This is to enable the
telescope to be revolved horizontally without re-levelling.
Level the instrument in one position. Revolve 180° horizon-
tally, and correct one half the movement of the bubble by the
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68 SURVEYING,
wye-adjustment and the other half by the levelling-screws.
Repeat for a check.
68. Relative Importance of Adjustments. — The first
adjustment is by far the most important. The second can
only enter in the work when the telescope is revolved slightly
from its true position in the wyes. Most modern levels have
some device for holding the telescope in its proper position
when in use. This position is such as brings the horizontal
wire truly horizontal. The last adjustment given is only a
matter of convenience. It saves stopping to relevel after re-
volving the telescope. It docs not affect the accuracy of the
work appreciably. It is absolutely essential, however, that the
line of sight should be truly horizontal when the bubble is in
the middle of the tube, or reads zero, and this makes the first
adjustment here given of such vital consequence.
69. Focussing and Parallax. — The eye-piece serves to
give a distinct and magnified view of the image. It also inverts
the image in all instruments where the object is seen in an
erect position. Since the magnifying power of the eye-piece
is large, its focal range of distinct vision is very small, depend-
ing on its magnifying power. With the ordinary field-instru-
ments it is about one sixteenth of an inch. Both the image,
as formed by the objective, and the cross-wires, should lie in
the focus of the eye-piece. They should therefore lie in the
same plane. Now the image may be moved back and forth
by moving the objective in or out, but the plane of the cross-
wires is fixed. If the two are brought into the same plane,
therefore, the image must be brought upon the wires. To
accomplish this, first fpcus the eye-piece on the wires so that
they appear most distinct. In doing this there should be no
image visible, so that either the objective is thrown out of
focus or the telescope is turned to the sky. The eye-piece is
most accurately focussed by finding its inner and outer limits
for distinct vision of the wires, and then setting it at the mean
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ADJUSTMENT, USE, AND CARE OF INSTRUMEXTS, 69
position. The objective may now be moved until the image
also comes into focus. This will have to be done for each
pointing if the distances are different. If the image is not
brought into exact coincidence with the cross-hairs, these will
seem to move slightly on the image as the eye is moved behind
the eye-piece. This angular displacement of the wires on the
image is called /ara/Zj^jr, and can only occur when they are not
in the same plane. It is removed by refocussing the object-
ive, thus moving the image, until there is no perceptible rela-
tive movement of wires and image as the eye is shifted, when
they are practically in coincidence. If there 'is parallax, the
reading may be in error by its maximum angular amount. If
the eye were always held at the centre of the eye-piece there
would be no parallax, and it is to accomplish this that the eye-
piece is covered by a shield with a small hole in its centre.
Still, the slight movement of the eye thus allowed is sufficient
to cause some parallactic error if the wires and image are not
practically coincident. When the eye-piece is once adjusted to
distinct vision on the cross-wires it requires no further atten-
tion so long as the instrument is used by the same person.
Another person, having eyes of a different focal range, would
have to readjust the eye-piece. The eye-piece adjustment,
therefore, is personal, and is made .once for all for a given indi-
vidual; while the objective adjustment depends on the dis.
tance of the object from the instrument, is made for each
pointing, and is considered perfect when the parallax is re-
moved.*
* This discussion is worded for an erecting telescope, where the objective
moves. In an inverting instrument the eye piece and reticule may move togethtf
in the telescope while the objective remains fixed. Here the image takes differ
cnt positions in the telescope-tube, as the distances vary, and the cross-wires
are moved to suit. There is also a motion of the eye-piece with reference to
the wires, and this is the eye-piece adjustment ; while the movement of both
together is what is called the objective adjustment in the above discussion.
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Fig. 13a.
69a. Architect's Compass Level. — Fig. 13^ is a cut of a
cheap but very useful instrument known as an Architect's
Compass Level.* It is a combination of a level and a needle
compass, and is used for laying out buildings, running ditches,
street grades, and especially for obtaining both a plan and pro-
file of a line by once running it. A great deal of work was
done on the Mississippi River Survey with an improvised in-
strument of this kind, in running trans-alluvial level lines from
bluff to bluff across the bottom lands subject to overflow. A
similar instrument, without the compass-box, but having the
graduated circle, reading by vernier to five minutes of arc, is
manufactured by several instrument-makers. It is called an
architect's level, and is very generally used by architects and by
surveyors in rural practice. These instruments cost only about
one-half as much as the standard engineer's level. They have no
clamp and tangent screws, but this is not a serious objection.
* Manufactured by Queen & Co., of Philadelphia, Pa.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 6gd
THE LEVELLING-ROD.
70. The Levelling-rod is used to measure the vertical dis-
tance from the line of sight down to the turning-point of
bench-mark. There are two general classes, Self-reading, or
Speaking, and Target Rods.
A Self-reading, or Speaking, Rod is one so graduated as to
enable the observer to note at once the reading of the point
which lies in the line of sight, this reading being in all cases
the distance to the bottom of the rod. The rod-man here has
nothing to do but to hold the rod vertical. The observer
notes and records the reading.
A Target-rod \?> furnished with a sliding target moved by the
rod-man in response to signals from the ob-
server until it accurately coincides with the
line of sight. Its position is then read with
great accuracy by means of a vernier scale.
Fig. 14 is one form of self-reading rod
which is also fitted with a target. This is
called the Philadelphia rod. Fig. \^a is the
New York rod, and is not self-reading. It is
the standard target-rod used in this country.
The one here shown is in three sections,
whereas those in common use are in two
parts only.
It is necessary that the rod be held vertical
when in use, and on sloping ground, or when
the wind is blowing, it is difficult to do this.
To insure a vertical rod, therefore, especially
in the plane of the line of sight, two level-
bubbles are sometimes attached, such as shown in the accom-
panying cuts. When not in use they can be removed and
folded up as shown.
Another method of attaining the same end is by means of
Thompson's Levelling Target, shown in Fig. \^a. This- target
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i^ic. 14.
CJ,_
Fia 15.
Fig. Z5a.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 7\
is bent at right angles, and so lies against two faces of the rod.
If held so that both faces show, the middle dividing line will
appear as a broken line when the rod is not vertical.
Most targets slide on the rod, and have a clamp screw and
springs. •When the rod is wet, the target is apt to stick and
move with a jerking motion. The target shown in Fig. 15 is
mounted on rollers in order to obviate this difficulty.*
Various patterns of self-reading rods are used. For rough
work a twelve- or fourteen-foot rod, 2 inches wide and i^ inches
thick, painted and fitted with an iron or brass shoe at bottom,
graduated to hundredths of a foot, will be found very efficient.
The graduations should be so distinct that they can be read
through the telescope at a distance of five or six hundred feet.
THE USE OF THE LEVEL.
71. The Level is used—
{a) To find the relative elevation of points a considerable
distance apart.
{b) To obtain the profile of a line.
{c) To establish a grade.
These objects may be more or less intermingled in any
given piece of work. Whatever may be the ultimate object of
the work, however, the immediate object for any given setting
of the instrument is to find how much higher or lower a certain
forward, or unknown, point is than a certain other back, or
known, point. Thus, the rod being held on the known point,
the line of sight is turned upon it and the rod-reading gives at
once the height of instrument above that point. If the rod be
now held on the forward, or unknown, point, and the line of
sight turned upon it, this rod-reading gives the distance of that •
point below the line of sight. The reading on the known
point is called the back-sight^ and that on the unknown point
is called the fore-sight. If the elevation of the known point
be given, we find the elevation of the line of sight by adding
* Both these targets are manufactured by Keufel & Esser, New York.
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72 SURVEYING.
the rod-reading at that point. By subtracting from this eleva*
tion the reading on the unknown point, the elevation of that
point is obtained. Thus we have found the relative elevations
of the two points by referring them both to the horizontal
plane through the instrument. Since the back-sigHl reading
gives the elevation of the instrument, and since this is always
greater than the elevation of that point, it follows that the
back-sight reading is essentially positive. For a similar reason
the foiesight reading is essentially negative, since any point
on which the rod is held is lower than the line of sight.
It will" also be seen that there can be but one back-sight (un-
less the height of the instrument is to be found from readings
on several known points, and the mean taken), while there can
be any number of fore-sights from one instrument position.
Thus, the height of the instrument having been determined,
the elevations of any number of points, in any direction, may
be determined by referring them all to the horizontal plane
through the instrument, whose elevation has been obtained by
the single back-sight reading. It is also important to remem-
ber that the terms " back-sight" and ** fore-sight" have no
reference to directions or points of the compass, but they do
have a rational significance when we think of the work pro-
ceeding from the known point to the unknown point or points.
Thus, we refer back to the known point for height of instru-
ment, and then transfer this knowledge forward to the points
whose elevations we wish to find.
DIFFERENTIAL LEVELLING.
72. DifTerential Levelling^ consists in finding the differ-
ence of elevation of points a considerable distance apart. The
elevation of the first point being known or assumed, the differ-
ence of elevation between this and any other point is found
and added algebraically, thus giving the elevation of the second
point. The ** plane of reference" is the surface of zero-eleva-
tion and is generally called the '* datum plane/* This is not
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 73
really a plane but a level surface, according to the definition
given in art. 60. It is, however, universally denominated the
'•plane of reference," "datum plane," or simply "datum."
The problem, then, is to find the difference of elevation between
two distant points. If the points were near together and had
not too great a difference of elevation, a single setting of the
instrument would be sufficient. If they are too far apart for
this, either in distance or in elevation, then more than one
setting of the instrument must be made. In this case the
intervening points occupied by the rod are called turning-points,
the terminal points being called bench-marks. The successive
differences of elevation of these turning-points is determined
by setting the level equally distant from them, and so they
serve to divide up the total distance between terminal points
into a series of short spaces, each of which can be covered by
a single setting of the instrument. The successive differences
of elevation of turning-points being found, their algebraic sum
would be the difference of elevation of the terminal points, or
bench-marks. But since all the back-sights are essentially
positive and all the fore-sights are essentially negative, we may
at once add all the back-sights together and all the fore-sights
together, and take the difference of the sums. This is the
difference of elevation between terminal points, and has the
sign of the larger sum, the back-sights being positive and the
fore-sights negative. This difference of elevation added alge-
braically to the elevation of the initial point gives the elevation
of the final point. Evidently the route travelled in passing
from one bench mark to another is of no consequence so long
as the true difference of elevation is obtained.
73. Length of Sights. — Where the ground is nearly level
it is desirable to make the length of sights (distance from
instrument to rod) as long as practicable, in order to increase
the rate of progress. For the best work this distance may be
from 100 to 300 feet, according to the state of the atmosphere.
When the air and ground differ greatly in temperature there
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74 SURVEYING.
result innumerable little upward and downward currents of
air, the upward being warmer than the downward currents.
The warmer air is more rarefied than the colder, and thus a
ray of light passing from the rod to the instrument passes alter-
nately through denser and rarer media, each change producing
a slight refraction of the ray. This causes a peculiar tremulous
condition of the image in the telescope, so that it is difficult to
determine just what part of it is covered by the cross-hairs. At
such times the air is said to be " trembling** or ** dancing'* or
** unsteady.** It always occurs more or less in clear weather,
owing to the earth then being hotter than the air, and it varies
with the quality of the soil, cinders or gravel being very bad.
When the air is in this condition the length of sights should
be shortened.
The back and fore sights for any setting of the instrument
should always be equal in length. Levelling is the only kind of
field-surveying wherein the instrumental errors may be thor-
oughly eliminated without duplicating the observations. This
may be done in levelling by making the back and fore sights
of equal length. For, since the dijference between back and
fore sights is always the quantity used, it follows that if both
are too large or too small by the same amount, the dijference
will be unchanged. If, when the bubble is in the middle of
its tube, the line of sight is inclined upwards by a given small
angle, then it has this relation to the horizontal on both fore
and back sights, and if the lengths of sights were equal the fore
and back rod-readings were equally in error. It is therefore
very desirable that these sights should be made of equal length.
Moreover, the effect of the earth's curvature is eliminated by
so doing, however long the sights may be. There are other
kinds of errors that are not eliminated by this means, but those
that are eliminated are of sufficient importance to warrant
great care to secure equal sights for each setting. If it is impos-
sible to do this at any time, the inequality should be balanced
off at the next one or two settings, by making them unequal
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ADJUSTMENT. USE, AND CARE OF INSTRUMENTS, 7$
in the opposite direction by the same amount. The equality
of sights can be determined by pacing with sufficient accuracy.
74* Bench-marks are fixed points of more or less perma-
nent character whose elevations are determined and recorded
for future reference. The general and particular location of a
bench-mark should be so distinctly described that any one
could find it from its description. Whenever the work is tem-
porarily interrupted a temporary bench-mark is set, such as
a substantial stake driven into the ground, or a spike in the
root of a tree. The prime requisite of a good bench-mark is
that it shall not change its elevation during the period in
which it is to be used. If this period is not more than two or
three years, a spike driven in the spreading root of a tree
near the trunk and well above ground will serve. The wood
should be trimmed away from it so as to leave a projecting
spur that will not be overgrown. The tree itself should then
be marked by notching or otherwise, and carefully located in
the description.
If the mark is to serve for from five to fifty years, stone or
brick structures or natural rock should be selected. The water-
tables, or corners of stone steps, of buildings, copings of founda-
tion and retaining walls, piers and abutments of bridges, or
copper bolts leaded in natural rock may serve. If artificial
structures are chosen, those should be selected which have
probably settled to a fixed position, and for this reason old
structures are preferable to new ones.
When stakes are used for temporary benches it is often
advisable *^^o set two or even three for a check. In this case
the mean elevation is the elevation used. In starting from
such a series of benches there would be as many back-sights
for the first setting of the instrument as there were benches,
the mean of which, added to the mean elevation of the benches,
would give the height of instrument. In running a continuous
line of levels it is advisable to set a benchmark at least as
often as one to the mile.
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76
SURVEYING.
75. The Record in differential levelling is very simple.
The bubble always being put in the middle of the tube, and
the rod-positions chosen equally distant from the instrument,
the bubble-reading and the length of sights may be omitted
from the record, unless some knowledge of the distance run is
desired, when the length of sights may be inserted.
Form of Record for Differential Levelling.
No. of
SlaUon.
Back-sights.
Forc-sigbts.
BlevatioD of
Mean Benches.
Remarks.
3.426
3.878
4.879
3-472
96.301
94.718
B. S. on B. M. 31
*. .. .. 3, a
I
2
3.652
4.517
3.216
3
4.361
4.873
F. S. on B. M 32
*. ». M 33a
4617
+11.385
— 12.968
+11.385
- 1.583
-
It will be seen that tRe mean of the readings on the two
bench-marks was used in each case. The back-sights being
essentially positive and the fore-sights essentially negative,
these signs are prefixed to the sums, and the algebraic sum of
these gives the elevation of the forward above or below the
rear benches. This added to the elevation of the initial point
gives the elevation of the final point. These points are the
mean elevation of two bench-marks in the example given.
76. The Field-work should be done with great care if
the best results are to be obtained. The instrument should be
adjusted every day, especially the parallelism of bubble-axis
and line of sight. The instrument and rod should both be set
in firm ground. An iron pin, about one inch square at top
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ADJUSTMEST, USE, AND CARE OF INSTRUMENTS. 77
six to eight inches long, and tapering to a point, should be
used for the turning-point. A rope or leather noose should be
passed through an eye at top to serve as a handle. To hold
the rod upright the rodman should stand squarely behind it
and keep it balanced on the pin. When the target is set and
clamped the rodman reads it and records it on a paper he car-
ries for the purpose. He then carries it to the observer, if it
was a back-sight reading, or he awaits the coming of the ob-
server if it was a fore-sight reading, when the observer also
reads it and records it in his note book. The rodman then
calls off his reading, and the observer notes its agreement with
his recorded reading. In this way two wholly independent
readings are obtained and any erroneous reading corrected.
Errors of one foot or one tenth are not very uncommon in
reading target rods. The rodman should be especially careful
to protect the turning-point from all disturbances between the
forward and back readings upon it. The observer must not
only obtain an accurate bisection on the target, but he must
know that the bubble is accurately in the centre of the tube
when this bisection is obtained. When the observer walks for-
ward to set his instrument he counts his paces, and takes as
long a sight as the nature of the ground or the condition of
the atmosphere will allow. When the rodman comes up he
counts his paces to the instrument and then goes t lie same dis-
tance in advance. Thus the observer controls the length of
sights, making them whatever he likes ; and it is the business
of the rodman to see that the back- and fore-sight for every
instrument-station are equal.
PROFILE LKVELLING.
77. In Profile Levelling the object is to obtain a profile of
the surface of the ground on certain established lines. Here
both the distances from, and the elevation above, some fixed
initial point must be obtained. When the line is laid out
stakes are usually driven every hundred feet, thCBe positions
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78 SURVEYING,
being obtained by a chain or tape. It is now the business of
the leveller to obtain the elevation of the ground at each of
these stakes, and at as many other intermediate points as may
be necessary to enable him to draw a fairly accurate profile of
the ground. The lOO-foot stakes are usually numbered, and
these numbers are entered on the level record. The inter-
mediate points are called pluses. Thus, a point 40 feet beyond
the twenty-fifth 100-foot stake is called 25 + 40> being really
2540 feet from the initial point. It is evident that no plus-
distance can be more than 100 feet, and these are usually paced
by the rodman. The intermediate points are selected with
reference to their value in determining the profile. These are
points where the slope changes, being mostly maximum and
minimum points, or the tops of ridges and bottoms of hollows.
Turning-points are selected at proper distances, depending on
the accuracy required, and these may or may not be points in
the line whose profile is desired. The levelling-instrument also
is not set on line, if it is found more convenient to set it off the
line.
In profile levelling, since absolute elevations with reference
to the datum-plane are to be obtained from every instrument-
position, it is necessary to find the height of instrument above
datum for every setting, and from this height of instrument,
obtained by a single back-sight reading on the last turning-
point, the elevations of any number of points are found by sub-
tracting the readings upon them.
78. The Record in profile levelling is much more elaborate
than in differential levelling. The following form is considered
very convenient for profile-work where the line has been laid
out and lOO-foot stakes set :*
*This sample page was contributed lo Engineering News in June, 1879, and
the form of record is credited to Mr. E. S. Walters, a railroad engineer of large
ex^rience.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 79
Guatemala and Honolulu Railroad. Feb. 30, 1876.
B. s.
El. of T. P.
aDd B. M.
F. S.
H.I.
206.049
I. S.
S E.
Sta.
B. M.
188 + 44
189
190
+ 30
191
+ 20
192
T. P.
193
+ 50
B. M.
194
195
T. P.
196
--65
198
+35
200
201
Remarks.
10.552
195-497
9.32
II. 41
7.01
2.07
1.62
0.38
0 82
196.73
194.64
199.04
203.98
204.43
205.67
205.13
0.515
202.797
3.252
203.312
3-10
2.70
5.264
8.20
9-35
200.21
200.61
195.11
193.96
198.048
3.411
194.840
8.472
198.251
4.28
5.06
7.20
10.60
7.00
5-46
193-97
193.19
191-05
187.65
191.25
192.79
9.527
195.083 t 3.168
204.610
10.25
8.62
6.04
194.36
195.99
198.57
24.005
14.892
14.892
9'"3
1
In the above headings, B. S. denotes back-sight; F. S., fore-sight ; I. S.,
intermediate sight ; H. I., height of instrument ; T. P., turning-point ; B. M.,
bench-mark ; S. E., surface-elevation ; Sta., station.
It will be noted that there is but one back-sight and one
height of instrument for each setting. The back-sight and
fore-sight readings from the same instrument-station are not
found here on the same line, as in differential levelling, but the
fore- and back-readings on the same turning-point are on the
same line. Thus, the rod was first read on t^ bench-mark
whose elevation .was known to bet i9S497^fe4fcibove datum.
The reading on this bench was 10.552, thus|^Bkga height of
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8o SURVEYING,
instrument of 206.049. This is marked B. M. in the station
column, and evidently has but one reading upon it in starting
the work from it. A series of intermediate sights are then
taken at various lOO-foot stakes and pluses, the readings on
which, when subtracted from the H. I., give the surface-eleva-
tions at those points. When the work lias progressed as far in
front of the instrument as the B. M. was back of it, a turning-
point is set, and the reading upon it recorded in the column of
fore-sights. This reading was 3.252, which, subtracted from the
H. I. 206.049, gives 202.797 as the elevation of the turning-
point. The instrument is now moved forward and a back-
sight reading taken upon this T. P. of 0.515, which added to
202.797 gives 203.312 as the new H. I. At this setting a new
bench was established by taking an intermediate sight upon it
of 5.264, and writing the elevation in the B. M. column instead
of in the S. E. column. The readings on bench-marks and
turning-points are made to thousandths, while the intermediate
sights for surface-elevation are read only to hundredths of a
foot. The last height of instrument is checked by adding the
back-sights and fore-sights, taking the difference and applying
it to the elevation of the initial point with its proper sign, re-
membering that back-sights are positive and fore-sights negative.
The profile is now constructed by the data found in the S. E.
and Sta. columns, these being adjacent to each other. One
of the great merits of this form of record is that wherever
it is necessary to combine any two numbers by addition or
subtraction, they are found in adjacent columns. In construct-
ing the profile, some kind of profile or cross-section paper is
used, and the horizontal scale made much smaller than the
vertical. Thus, if the horizontal scale were 400 feet to the
inch, the vertical scale might be 10 or 20 feet to the inch.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 8 1
LEVELLING FOR FIXING A GRADE.
79. In fixing a grade the profile may be obtained and
the grade marked upon it. The vertical distance between the
surface-line and the grade-line, at any point is the depth of
cut or fill at that point, and this may be marked on the line
stakes at once, without the aid of the level or rod, if only the
centre depths are desired, as in the case of a ditch or trench.
If the sides are to have a required slope, however, the level
and rod are necessary to fix the horizontal distance of the
limiting or " slope" stakes from the centre stakes whenever
the ground is not strictly a level surface. This operation is
called '* cross-sectioning,*' and is described in Chapter XIII.,
on Determination of Volumes.
If the grade be known before the profile is determined, to-
gether with the absolute elevation of the initial point, as is
sometimes the case with ditches and trenches for pipe lines or
sewers, then the depth of cut (or fill) may be at once deter-
mined and marked on the line stakes when the profile is taken.
The form of record might be the same as given above for pro-
file levelling, with the addition of two columns after the " Sta-
tion" column, one being Elevation of Grade, and the other Cut
or Fill. The elevation of grade would be found for each pro-
file point by adding if an up, and subtracting if a down, grade,
the differences of elevation corresponding to the successive
distances in the profile. The difference between the corre-
sponding "surface-elevation" and "elevation of grade" would
be the cut or fill at each point, which could be at once taken
out and marked on the line stake.
THE HAND-LEVEL.
80. Locke's Hand-level is a very convenient little instru-
ment for rough work, such as is done on reconnaissance expedi-
tions. It consists of a telescope with a bubble attached in
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82 SURVEYhXC,
such a way that the position of the bubble is seen by looking
through the telescope. A horizontal line of sight is thus
readily determined. It is supposed to be adjusted once for all.
Fig. z6.
EXERCISES WITH THE LEVEL.
81. Adjust the bubble to the line of sight by the first, or indirect, method,
and then lest it by the second, or direct, method. If this second method does
not show it to be in adjustment, where does the error lie ?
82. Cause the line of sight and bubble-axis to make a considerable angle
with each other (that is, put it badly out of adjustment in this particular), and
level around a block or two, closing on the starting-point, being careful to
make back and fore sights as nearly equal as possible. Of course the final
elevation of the point should agree with the assumed initial elevation. The
difference of these elevations is the error of closure of the level polygon. If
the back and fore sights were exactly equal this should be zero, notwithstand-
ing the erroneous adjustment.
83. Put the instrument in accurate adjustment, and level over the same
polygon as before, making the back and fore sights quite unequal, and note the
error of closure. If the instrument were in exact adjustment and there were
no errors of observation, should the error of closure be zero?
84. Range out a line on uneven ground about a half-mile in length, and set
stakes every hundred feet. Let each student determine the profile indepen-
dently. When all have finished, let them copy their profiles on the same piece
of tracing-cloth, starting at a common point. The vertical scale should be
large, so as to scatter the several profile lines sufficiently on the tracing. Each
profile should be in a diflferent color or character of line.
85. Select a line on nearly level ground, about a half-mile in length. Estab-
lish a substantial bench-mark at each end. Let each student determine the
difference of elevation of these benches twice, running forward and back. See
if the results are affected by the direction in which the line is run.
If each student could do this several times some evidence would be ob-
tained as to there being such a thing as " personal equation" in levelling ; that
is, each person tending to always obtain results too high or too low. Why U
it improbable that there could be any personal equation in levelling?
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 83
CHAPTER IV,
INSTRUMENTS FOR MEASURING ANGLES.
THE TRANSIT.
86. The Engineer's Transit is the most useful and
universal of all surveying-instruments. Besides measuring
horizontal and vertical angles it will read distances by means
of s^dia wires, determine bearings by means of the magnetic
needle, do the work of a solar compass by means of a special
attachment, and do levelling by means of a bubble attached
to the telescope. It is therefore competent to perform all the
kinds of service rendered by any of the instruments heretofore
described, and is sometimes called the "universal instrument.*'
A cut of this instrument is shown in Fig. 17. Fig. 18 is a
sectional view through the axis of a transit of different
manufacture.
The telescope, needle-circle, and vernier plates are rigidly
attached to the inner spindle which turns in the socket G
Fig. 18. This portion of the instrument is called the alidade,
as it is the part to which the line of sight is attached. The
socket C carries the horizontal limb, shown at B, and may
itself revolve in the outer socket attached to the levelling-head.
Either or both of these connections may be made rigid by
means of proper clamping devices. If the horizontal limb B
be clamped rigidly to the levelling-head and the alidade spindle
be allowed to revolve, then horizontal angles may be read by
noting the vernier-readings on the fixed horizontal limb for
the different pointings of telescope. If the horizontal limb
itself be set and clamped so that one of the verniers- reads zero
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84
SURVEYING.
Fig, x».
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, Ss
when the telescope is on the meridian, then for any other
pointing of the telescope the reading of this same vernier
gives the true azimuth of the line. It is necessary, therefore,
to have two independent movements of telescope and horizon-
tal limb on the same vertical axis. The magnetic needle is
shown at N. The plumb-line is attached at P\ this should
always be in the vertical line passing through the centre of
the graduated horizontal circle. This will be the case when
it is attached directly to the axis itself, for this must always
be made vertical.
The limb is graduated from zero to 360°, and sometimes
with a second set of figures to 90° or 180°. There are two
verniers reading on the horizontal limb 180° apart. Both the
instruments shown in Figs. 17 and 18 have shifting centres,
enabling the final adjustment of the instrument over a point
to be made by moving it on the tripod-head. The telescope
is shorter than those used in levelling-instruments in order
that it may be revolved on its horizontal axis without having
the standards too high. It is called a transit instrument on
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S6 SURVEYING,
account of this movement, which is similar to that of an
astronomical transit used for observing the passage (transit)
of stars across any portion of the celestial meridian. When
the telescope is too long to be revolved in this way the instru-
ment is called a theodolite. This is the only essential differ-
ence between them.**^ The ** plain transit *' has neither a
vertical circle nor a bubble attached to the telescope.
ADJUSTMENTS OF THE TRANSIT.
87. The Adjustments of the Engineer's Transit are
such as to cause '(i) the instrument to revolve in a horizontal
plane about a vertical axis, (2) the line of coUimation to gen-
erate a vertical plane through the instrument-axis when the
telescope is revolved on its horizontal axis, (3) the axis of the
telescope-bubble to be parallel to the line of collimation, thus
enabling the instrument to do levelling, and (4) the vernier on
the vertical circle so adjusted that its readings shall be the
true altitude of the line of collimation. These four results are
attained by making the following five adjustments :
88. First. To make the Plane of the Plate-bubbles
perpendicular to the Vertical Axis. — This adjustment is
the same as with the compass. (One of the plate-bubbles is
usually set on one pair of standards.) Bring both bubbles to
the centre, revolve 180°, correct one half the movement on
the levelling-screws and the other half by raising or lowering
the adjustable end of the bubble-tube. Each bubble should
be brought parallel to a set of opposite levelling-screws in
making this adjustment, so that the correcting for one bubble
does not throw the other out. When either bubble will main-
tain a fixed position in its tube as the instrument is revolved
horizontally, the axis of revolution is vertical. One bubble is
*The first engineer's transit instrument was made by Wm. J. Young (now
Young & Sons), Philadelphia, 1831. All American engineer's altitude-azimuth
instruments are now made to revolve in this way
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 87
therefore sufficient for making this axis vertical, but two are
somewhat more convenient, especially for indicating when the
axis has become inclined from unequal settling or expansion
while in use.
89. Second. To make the Line of Sight perpendicular
to the Horizontal Axis of the Telescope.* — When this is
done, the line of sight will generate a plane when the tele-
scope is revolved on its horizontal axis. If the line of sight
is not perpendicular to the horizontal axis, it generates the
surface of a cone when the telescope is revolved, the axis of
the cone being the axis of revolution, and the apex being at
the intersection of the line of sight with this axis.
Set the instrument on nearly level ground, where a view
can be had in opposite directions. Set the line of sight on a
definite point a few hundred feet away. Revolve the telescope
and set another point in the opposite direction. Revolve the
alidade until the line of sight comes upon the first point. Re-
volve the telescope again and fix a third point on the line of
sight beside the second point set. Measure off one^ourth the
distance between these two points from the last point set, and
bring the line of sight to this position by moving the reticule
laterally. This movement of the reticule is direct in an erect-
ing instrument and reversed in an inverting instrument.
The student should illustrate the correctness of this method
by means of a figure. The four pointings were the intersec-
tions of a diametral horizontal plane with the surfaces of the
the two cones generated. These cones were pointed in oppo-
site directions, but had one element in common, being the two
pointings to the first point. The two opposite elements
diverged by four times the difference between the semi-angle
of the cone (subtended by the line of sight and the axis of
rotation) and 90°.
90. Third. To make the Horizontal Axis of the Tele-
* This is called the Adjustment for Collimation, since it consists in bringing
the line of sight into coincidence with the line of collimation, which is simply the
true position for the line of sight.
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68 SURVEYING.
scope perpendicular to the Axis of the Instrument.— When
this is done the former is horizontal when the latter is vertical,
and, the second adjustment having been made, the line of sight
will generate a vertical plane when the telescope is revolved.
Set the instrument firmly and level it carefully. Suspend
a plumb-line some 20 or 30 feet long, some 15 or 20 feet from
the instrument. The weight should rest in a pail of water and
the string should be hung from a rigid support. There should
be no wind, and the cord should be small and smooth. A small
fish-line is very good. Care must be exercised that the weight
does not touch the bottom of the pail from the stretching of
the cord. Set the line of sight carefully on the cord at top,
the plate-bubbles indicating a strictly vertical instrument-axis.
Clamp both horizontal motions and bring the telescope to read
on the bottom portion of the cord. The cord is apt to swing
to and fro slightly, but its mean position can be chosen. If the
line of sight does not correspond to this mean position, raise
or lower the adjustable end of the horizontal axis until this
test shows the line of sight to revolve in a vertical plane.
Constant attention must be given to the plate-bubbles to see
that they do not indicate an inclined vertical axis.
Or, two points nearly in a vertical line may be used, as the
top and bottom of the vertical corner of a building. Set on
the top point and revolve to the bottom point. Note the
relation of the line of sight to this point. Revolve 1 80° about
both vertical and horizontal axes, and set again on the top
point. Lower the telescope again and read on the bottom
point. If the telescope-axis of revolution is horizontal, the
second pointing at bottom should coincide with the first. If
not, adjust for one half the difference between these two
bottom readings. \
It will be noted that the second and third adjustments are
necessary to the accomplishment of the second result cited in
art. 87.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 89
91. Fourth. To make the Axis of the Telescope-bub-
ble parallel to the Line of Sight. — This adjustment is per-
formed by means of the ** peg-adjustment/* as described in
art. 65, p. 65, second method. The height of the instrument
may now be measured to the centre of the horizontal axis if it
be found more convenient than sighting backwards through
the telescope. When this adjustment is made the instrument
is competent to do levelling the same as the levelling-instru-
ment. The telescope is not quite so stable, however, in the
transit because it is mounted on an axis instead of in two rigid
wyes.
92. Fifth, To make the Vernier of the Vertical Circle
read Zero when the Line of Sight is Horizontal.— Having
made the axis of the telescope-bubble parallel to the line of
sight, bring this into the centre of its tube, and adjust the
vernier of the vertical circle till it reads zero on the limb. If
this vernier is not adjustable, the reading in this position is its
index error. The line of sight might still be adjusted to the
vernier by moving the reticule, and then adjusting the bubble
to the line of sight. To do this use the ** peg-adjustment '* as
described in art. 65, making the vertical circle read zero each
time, and paying no attention to the telescope-bubble. Correct
the line of sight by ^ d. as given by Eq. (2), p. 66, by moving
the reticule, and this should give a horizontal pointing for a
zero-reading of the vertical circle. Then adjust the bubble to
this reading by bringing it to the centre of the tube by means
of the vertical motion at one end of the bubble-tube. If the
reticule is disturbed after making the second adjustment, that
adjustment should be tested again to see if it had been dis-
turbed.
93. Relative Importance of the Adjustments.— The first
adjustment is important in all horizontal and vertical angular
measurements. In measuring vertical angles the error may be
the full amount of the deviation of the vertical axis from the
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90 SURVEYING,
vertical, and in measuring horizontal angles something very
much less than this.
The second adjustment is more important in the running of
a straight line by revolving the telescope than in any other kind
of work, for here the error in the continuation of the line is
twice the error of adjustment. It is also important in measur-
ing horizontal angles between points not in the same horizontal
plane.
The third adjustment is most important in the measure-
ment of horizontal angles between points not in the same hori-
zontal plane, as in the determination of the azimuth of a line
by an observation on a circumpolar star.
The fourth and fifth adjustments are important only in
levelling operations, either by reading the vertical angle or by
the use of the bubble.
INSTRUMENTAL CONDITIONS AFFECTING THE ACCURATE
MEASUREMENT OF HORIZONTAL ANGLES.*
94. Eccentricity.— This is of two kinds: (i) eccentricity of
centres, and (2) eccentricity of verniers. If the axis of the coni-
cal outer socket C, Fig. 18, is not exactly in the centre of the
graduated limb -5, then when the telescope with the vernier
plates Fare revolved in this socket, the verniers will have an
eccentric motion with reference to the graduated limb. If the
line joining the zeros of the verniers passes through the axis
of the socket, it is evident that there is but one position of
these verniers which will give readings on the limb 180° apart,
and that is when both centres lie in this diametral line. For
all other positions of the verniers, one of them will read as
much too large as the other does too small ; so that if the mean
* For extended discussions of this subject, see Bauernfeind*s " Vermessungs-
kunde," § 144, vol. i., and Jordan's *' Handbuch der Vermessungskunde/' §88,
vol. i. Also translations from these, by Prof. Eisenmann, in Journal of the
Association of Engineering Societies, vol. iv. p. 196.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, QI
of the two vernier-readings be taken, this error from eccentric-
ity would be eliminated.
Eccentricity of verniers is due to their zeros not falling on
a diametral line through the axis of the spindle; in other
words, they are not 1 80° apart. This involves no error in
measuring horizontal angles. It is convenient, however, to
have the verniers read exactly 180** apart. In any case, read-
ing of both verniers and taking the mean eliminates all errors
from eccentricity. An eccentricity of centres of one one-thou-
sandth of an inch would cause a maximum error of i'-o8' on a
six-inch circle if but one vernier were read. It is not unusual
for an instrument to have an eccentricity of centres of several
times this amount, either from wear or from faulty construction,
or both. The necessity for reading both verniers in all good
work is therefore apparent.
95. Inclination of Vertical Axis. — ^The horizontal angle
between points at different elevations is obtained by measuring
the horizontal angle subtended by two vertical planes passing
through these points and the point of observation. These
vertical planes are the planes described by the line of sight as
the telescope is revolved. By this means the points may be
said to be projected vertically on the horizontal plane and
then the angle measured. If the vertical axis of the instru-
ment is somewhat inclined, these projecting planes are not ver-
tical, neither do they have the same inclination to the horizon
on different parts of the limb. The projecting planes through
two points will therefore neither be vertical nor equally in-
clined to the horizon. The measured horizontal angle thus
obtained will therefore be in error. The vertical axis is always
inclined when the plate-bubbles are not in adjustment or when
they do not show a level position.
If the axis be inclined 5' from the vertical, and leadings be
taken on points 60° apart, one being 10° above and the other
lO** below the horizon, the maximun error from this source
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92 SURVEYING.
would be about i'. If the inclination in this case were i°, the
maximum error would be i8'. This shows the importance of
keeping the plate-levels in adjustment and of watching them
during the progress of the work to see that they remain in the
centre.
96. Inclination of Horizontal Axis of Telescope.—
This causes the plane generated by the line of sight to be in-
clined from the vertical as much as the axis of -revolution is
from the horizontal. The projecting planes are therefore all
equally inclined, and the resulting error in horizontal angle is
a function of the difference of elevation of the two points. If
one point is 10® above and the other 10° below the horizon,
and if the inclination of the axis is 5', the resulting error in
the measurement of the horizontal angle is i'-45^ This error
is not a function of the size of the horizontal angle, and would
be the same for two points in the same vertical plane, the in-
strument indicating a horizontal angle of 1' 45'' between them
for the case here chosen. In making the adjustment of the
horizontal axis by means of the plumb-line, if the line be 15
feet distant and suspended 15 feet above the instrument, then
the pointing to the top will have an altitude of 45°. In this
case the angular error made in bisecting the plumb-line will be
the angular divergence of the axis of rotation from th^ hori-
zontal. If the combined error of the two bisections be o. 05 in.,
the angular error in the adjustment will be 1'. The adjust-
ment may readily be made closer than this.
Errors from this source are eliminated by revolving the
telescope and reading the same angle in the reversed position.
The mean of the two values will be independent of this error.
If many measurements are made of one angle, there should be
an equal number with telescope direct and reversed.
The student should show by a figure how this elimination is effected by the
reversal of the telescope.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 93
97. The Line of Sight not being Perpendicular to
the Horizontal Axis. — This causes the projecting planes
to be conical surfaces, which become vertical on the horizon.
Since the error of collimation is necessarily a small angle, thus
causing the conical surface to be very nearly a plane, and since
this surface is vertical on the horizon, the resulting error in
measuring horizontal angles is very small unless the difference
in the elevations of the points is very great. If the points are
distant, as they always are in the accurate measurement of
horizontal angles, then their angular elevation is necessarily
small, so that this source of error is insignificant in this kind
of work. When straight lines are prolonged by reversing the
telescope, however, this adjustment becomes very important,
for the error then enters the work with twice its angular
amount. It is eliminated by revolving the alidade until the
line of sight, with telescope reversed, falls again on the rear
point, and again revolving the telescope. The point now
falls as far on one side of the true position as it before did on
the other. The middle point lies therefore in the line pro-
longed.
Let the student illustrate by diagram.
THE USE OF THE TRANSIT.
98. To measure a Horizontal Angle. — Having centred
the instrument over the vertex of the angle required, take a
pointing to one of the points and clamp both alidade and
limb. Make the final bisection by means of either tangent-
screw. Read the two verniers, and record them, calling one
the reading of vernier A and the other of vernier B, Loosen
the alidade clamp and turn upon the second point, clamp, and
set by the upper tangent-screw. Read both verniers again.
Correct the readings of vernier A by half the difference be-
tween the A and B readings in each case. The difference
between these corrected readings is the value of the angle.
7
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94 SURVEYING.
Be careful not to disturb the lower clamp- or tangent-screw
after reading on the first point. If there are two abutting
tangent-screws for the lower plate, be sure that both are
snug, otherwise there may be some play here which would
allow the limb to shift its position, in which case the true angle
would not be obtained. If there is but a single tangent-screw
working against a spring on the other side of the armature,
as shown in Fig. i6, then there can be no lost motion unless
the friction on the axis is greater than the spring can over-
come, which should never be the case.
Do not set the clamp-screws too tightly, as it strains and
wears out the instrument unnecessarily. A very gentle press-
ure is usually sufficient to prevent slipping. This caution
applies equally well to all levelling-, adjusting-, and connecting-
screws in the instrument. The young observer is generally
inclined to set them up hard, as he would in heavy iron-work.
It must be remembered that brass is a soft material, easily dis-
torted and worn, and that the parts should be strained as little
as possible to insure against movement in ordinary handling.
The subject of measurement of horizontal angles is further
discussed in Chapter XIV., on Geodetic Surveying.
99. To measure a Vertical Angle.— Vertical angles are
usually referred to the horizon, and are angles of elevation or
depression above that plane. If the vernier on the vertical
circle has been properly adjusted (or its index error determined
in case it is not adjustable and the line of sight has not been
adjusted to it), then the altitude of a point is obtained at once
by turning the line of sight upon it and reading the vertical
angle. Special attention must here be given to the bubble
parallel to the vertical circle, for it is on this bubble that the
accuracy of the result wholly depends. If there is but one
vernier, it is designed to read both ways, as is shown in Figs. 5
or 6, p. 19. In this case errors of eccentricity cannot be elim-
inated.
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ADJUSTMENT, USE. AND CARE OF INSTRUMENTS, 95
To eliminate errors of adjustment of the plate-bubbles and
of the vernier on the vertical circle, revolve the alidade i8o°, re-
level, read the vertical angle again with telescope in a reversed
position, and take the mean. This can only be done in case
the vertical limb is a complete circle. In many instruments it is
but a half-circle or less, in which case this elimination cannot
be made. The accuracy of the adjustments alone can then
be relied on, and these must be frequently tested. If the plate-
bubble parallel to the vertical circle, the telescope-bubble, and
the vernier of the vertical circle have all been once accurately
adjusted, then when these bubbles are brought to a zero-read-
ing the vertical circle should also read zero. This test can
always be readily applied, and, though not an absolute check,
it is a very good one, inasmuch as two of these three adjust-
ments would have to be out by the same amount and in the
same direction to still agree with the third.
100. To run out a Straight Line.— The transit-instru-
ment is especially adapted to the prolongation of straight Hnes,
as long tangents on railroads, and yet it requires the most care-
ful work and much repetition to run a line that approximates
very closely to a straight line.
Having determined the direction which the line is to take
from the initial point, set accurately over this point, turn the
telescope in the given direction, and set a second point at a
convenient distance. These two points now determine the
line, and it remains to prolong it indefinitely over such uneven
ground as may lie in its course. The line, when established,
is to be the trace of a vertical plane through the first two
points on the surface of the ground. If the line of sight
always revolved in a vertical plane, and no errors were made
in handling the instrument and in setting the points, the
problem would be easily solved, but we may safely say that
the surface generated by the Hne of sight never is a vertical
plane. (The adjustments being never absolutely correct.)
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96 SURVEY IS^G.
This surface is a cone whose axis is not strictly horizontal, for
both the horizontal and vertical axes are somewhat inclined
from their true positions. It remains then so to make the
observations that all these errors of adjustment will be elimi-
nated. The following programme accomplishes this :
(i) Set accurately over the forward point, putting one pair
of levelling-screws in the line.
(2) Clamp the horizontal Hmb in any position.
(3) Level carefully, and turn upon the rear point.
(4) Relevel for the bubble that hes across the line.
(5) Make the bisection on the rear point, revolve the tele-
scope, and set a point in advance. This may be a tack in a
stake set with great care by making the bisection on a pencil
held vertically on the stake.
(6) Unclamp the alidade and revolve it about the vertical
axis till the telescope comes on the rear point.
(7) Relevel for the cross bubble again.
(8) Make the bisection on the rear point, revolve the tele-
scope again, and set a second point in advance beside the first
one. The mean of these two positions should lie in the verti-
cal plane through the two established points, whatever may be
its elevation, and regardless of small errors in the instrumental
adjustments. For the reversals of the telescope and alidade
eliminated the errors of collimation and horizontal axis, while
the relevelling eliminated the error due to the error of adjust-
ment of the plate-bubble. If this bubble were out of adjust-
ment the vertical axis inclined as much to one side for the first
setting as it did to the other side for the second setting.
This operation may be repeated for a check, or to further
eliminate errors of observation. The instrumental errors are
wholly eliminated by one set of observations, as above given.
It will be noted that this method is independent of the gradua-
tion of the limb. The only assumptions are that the instru-
ment and its adjustments are rigid during the reversal of the
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 97
telescope, and that the pivots of the horizontal axis are true
cylinders.
loi. Traversing. — A traverse, in surveying, is a series of
consecutive courses whose lengths and bearings, or azimuths,
have been determined. When a compass is used the bearing
of each course is determined by the needle independently of
that of the preceding course. When a transit is used and the
needle not read, the graduated circle of the instrument is
always oriented, or brought into the meridian, by taking a
back-sight to the preceding station. If the azimuth* of the
first course is known with reference to the meridian, the
azimuth of all subsequent courses may be at once determined
by properly orienting the limb of the instrument at the suc-
cessive stations. Thus, if the south point has a zero azimuth
the limb of the instrument should be oriented at each station,
so that when the telescope points south vernier A shall read
zero.
T)^^ forward azimuth of a line is its angular deviation from
the south point when measured at the rear station forward
along the line.
The back azimuth of a line is its angular deviation from the
south point at the forward station when measured from that
station back along the line.
The forward and back azimuth differ by i8o® plus or minus
the convergence of the meridians at the two extremities of the
line. If this hne is north and south it lies in the meridian,
and hence its forward and back azimuth differ by i8o°. When
the course has an easterly or westerly component, or, in other
words, when its extremities have different longitudes, the
divergence of the line from the meridian at one end differs
from its divergence from the meridian at the other by as much
* In this treatise azimuth is always reckoned from the south point in the
direction S.W.N. E. to 360**. The bearing of the line is thus given by its
numerical value alone, without the aid of letters.
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98 SURVEYING,
as these meridians differ from parallelism. This is inappre-
ciable on short lines, and hence in traversing the forward and
back azimuth will be considered as differing by i8o°.
The field-work proceeds as follows, so far as the transit is
concerned. Let it be assumed that from the initial point A
of the survey the true azimuth to some other point Z is given.
Let the stations h^ A,B, C, etc.
Set vernier A to read the known azimuth AZ, With the
alidade and limb clamped together, turn the telescope on Z
and clamp the limb, setting carefully by means of the lower
tangent-screw. If the alidade be now loosened and vernier A
made to read zero, the telescope would point south. Turn
the telescope on B by moving the alidade alone, and the read-
ing of vernier A gives the forward azimuth of the line AS,
Move the instrument to B and set vernier A to read the back
azimuth of AB^ which is found by adding i8o" to or subtract-
ing it from the forward azimuth, according as this was less or
more than i8o°. With alidade and limb clamped at this read-
ing, turn upon -^, clamp the Hmb and unclamp the alidade, and
the instrument is again properly oriented for reading directly
the true azimuth of any Hne from this station, as the line BCy
ter instance. In this manner a traverse may be run with the
transit, the field-notes showing the true azimuth of each course
without reduction. The lengths of the courses may be found
in any manner desired.
If preferred, the telescope may be revolved on its horizon-
fiil axis and vernier A left with its forward reading, for orient-
itig. Then revolve the telescope back to its normal position
and proceed with the work.*
fc« I .. , ^ — ^
* For a method of computing the coordinates of the courses, and the use ol
the traverse table, see chapter on Land Surveying.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 99
THE SOLAR ATTACHMENT.
102. The Solar Attachment is a device to be fastened to
the telescope axis of a transit-instrument, thus making a com-
bination that will do the work of a solar compass. One form
of this device is shown in Fig. 19.* The various spherical
functions concerned in the problem are also represented in this
figure by their several great circles. The polar axis, declination-
arc, and collimation-arm are the same here as in the solar com-
pass. The latitude-arc is here replaced by the vertical circle
of the transit, and the telescope gives the line of sight. The
adjustments and working of this attachment are so nearly iden-
tical with those of the solar compass that they will not be
repeated here. If the student has mastered the principles
involved in the use of the solar compass he will have no diffi-
culty in using the attachment.
Various forms of solar attachments have been invented, the
most recent and perhaps the most efficient of which is that
shown in Fig. 20, invented by G. N. Saegmuller in 1881. It
is manufactured by Fauth & Co., Washington, D. C, and by
Keuffel & Esser, New York. It consists simply of an auxiliary
telescope with bubble attached, having two motions at right
angles to each other. These motions are horizontal and verti-
cal when the main telescope, to which the attachment is rigidly
fastened, is horizontal. If the main telescope be put in the
meridian and elevated into the plane of the celestial equator,
however, then the vertical axis of the attachment also lies in
the meridian but points to the pole. It therefore becomes a
polar axis about which the auxiliary telescope may revolve. If
this telescopic line of sight be at right angles to the polar axis,
it will generate an equatorial plane. If the line of sight be in-
clined to this plane by an amount equal to and in the direction
of the sun's declination, then when revolved on its polar axis it
*From Gurley's Catalogue,
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P10.19,
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Fig. 2o.
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I02 SURVEYING,
would follow the sun's path in the heavens for the given day,
provided the sun did not change its declination during the day.
It only remains, therefore, to show how the latitude and decli-
nation angles may be set off in order that the competency of
this instrument to do the work of the solar compass may be-
come apparent.
To set off the declination-angle, turn the main telescope
down or up according as the declination is north or south, and
set the declination-angle on the vertical circle. Bring the
small telescope into the plane of the large one and revolve it
about its horizontal axis until its bubble comes to the centre
of its tube. The angle formed by the two telescopic lines of
sight is the declination-angle. Revolve the main telescope
until it has an altitude equal to the co-latitude of the place,
and clamp it in this position. With the vertical motions of
both telescopes clamped, and their lateral motions free, if the
line of sight of the small telescope can be brought upon the
sun the main telescope must lie in the meridian. The vertical
circle of the transit is thus seen to do the work of both the
latitude and declination arcs of the solar compass.
103. Adjustments of the Saegmuller Attachment. —
First, All the adjustments of the transit must be as perfect
as possible, but especially the plate and telescope bubbles, the
vernier of the vertical circle, and the transverse axis of the
telescope.
Second. To make the Polar Axis perpendicular to the Plane
of the Line of Collimation and Horizontal Axis of the Main
Telescope, — Carefully level the instrument and bring the teles-
cope-bubble to the middle of its tube. The line of sight and
horizontal axis of this telescope should now be horizontal, so
that the polar axis is to be made vertical. To test this, revolve
the auxiliary telescope about the polar axis, and see if the
bubble on the small telescope maintains a constant position.
If not, correct half the movement by means of the adjusting,
screws at the base of the small disk, and the other half by re-
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, IO3
volving the auxiliary telescope. These adjusting-screws are
exactly analogous to the levelling-screws of the main instru-
ment.
Third. To make tlie Line of Sight of the Small Telescope
parallel to the Axis of the Attaclud Bubble. — Make the large
telescope horizontal by bringing its attached* bubble to the
middle of its tube. Bring the small telescope in the same plane
and make it also horizontal by means of its bubble, clamping
its vertical motion. Measure the vertical distance between the
axes of the two telescopes, and lay off this distance on a piece of
paper by two plain horizontal lines. Set this paper up at a con-
venient distance from the instrument, and on about the same
level. Bring the line of sight of the large telescope on the lower
mark, and see if that of the small telescope falls on the upper
mark. If not, adjust its reticule until its line of sight come on
the upper mark. Revolve back to the horizontal to see if both
bubbles again come to the middle simultaneously.
When this adjustment is completed, there should be five
lines in the instrument parallel to each other when instrument
and telescopes are level, — viz., the axes of the two telescope-
bubbles and of the plate-bubble on the standards, and the two
lines of sight, — and, in addition, the vernier on the vertical
circle, should read zero.
The seven adjustments (five of the transit and two of the
attachment) must all be carefully made and frequently tested
if the best results are desired. When this is done, this attach-
ment will give the meridian to the nearest minute of arc, if ob-
servations be taken when the sun is more than one hour from
the horizon and two hours from the meridian. The advantages
of the Saegmuller attachment consist mainly in having a teles-
copic line of sight, and in the use of the vertical limb of the
transit for setting off the declination and co-latitude. The
effect of small errors in the latitude and declination angles, such
as may be due to errors in the adjustments, is shown by the
table, art. 54, p. 51.
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103a. Determination of the Meridian by Direct Solar
Observation.* — Knowing the latitude of the place of observa-
tion and the declination of the sun, the angle which a vertical
plane through the sun
makes with the meri-
d i a n plane, counting
from the North point,
may be found by simply
observing the sun's alti-
tude. Thus, in Fig. 20a,
let Z be the z e n i t h
point, P the pole, and
5 the sun. Then know-
ing the latitude of the
place, the declination,
and the altitude of the
sun, the three sides of
the spherical triangle
ZPS become known,
since these sides are respectively the co-latitude, the co-decli-
nation, and the co-altitude. Knowing these three sides of the
spherical triangle, the solid angle A, or PZS, may be computed
from the formula
Fig. 10a.
Cos i A
\/:
sin S sin (S — codec.)
sin coiat, sin coalt.
where 5 is one-half the sum of the three sides.
This kind of a solar observation for azimuth does not neces-
sitate the use of the solar attachment, but it does require either
a colored shield over the object glass, together with a prismatic
eye-piece, or in place of these an auxiliary disk or diaphragm,
mounted or held just back of the eye end. This disk may
* Added to the twelfth edition, 1896.
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ADJUSTMENT. USE, AND CARE OF INSTRUMENTS, IO3*
even be a plain sheet of white paper held in the hand some
four to six inches back of the eye-piece. By having the
cross-wires in good focus, and the objective focussed on the
sun, so as to give a clearly defined image, this image may be
brought centrally upon the shadow of the cross-wires by the
vertical and horizontal slow-motion screws, within the limits of
accuracy of reading the graduated circles. The reading of the
vertical circle gives the altitude of the sun, from which the
co-altitude is found.
The declination of the sun at the time of the observation is
obtained from the nautical almanac, the same as if it were to
be used with the solar attachment. The correction for refrac-
tion is also applied.* The time of the observation is recorded
and the declination of the sun at this time can be afterwards
determined. The horizontal circle is also read, and a pointing
nade to the azimuth mark, and the horizontal circle read again.
These observations can be repeated as often as desired ; prob-
ibly three sets of readings would usually be taken.
The time of day best suited to this observation, like that
for an observation with the solar attachment, is near the
middle of the sun's path from the horizon to the meridian,
either before or after noon. That is to say, the altitude of
the sun should be changing rapidly at the time of the obser-
vation. If taken too near the horizon the correction for
refraction becomes large and uncertain, and if taken too near
noon the altitude is changing too slowly to furnish a good
argument. It needs scarcely to be said that the transit must
be in as good adjustment for this kind of direct observation
as is required when the solar attachment is used. As in that
case also, the mean of two observations taken at symmetrical
times before and after noon will be free from the errors of
adjustment, and of the latitude and declination used. This
method is now commonly employed in the mining regions of
the West, where it is rapidly replacing the use of the solar
attachment.
♦ In this case the correction is added to the observed altitude, as given on
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SURVEYING,
The following detailed directions for making these observa-
tions were prepared by A. W. French, Instructor in Civil Engi-
neering in the Thayer School of Civil Engineering, Dartmouth
College : *
*' I. Observe the sun directly, by the aid of colored glass, and bring his image
tangent to the horizontal and vertical wires ; read the vertical circle and the hori-
zontal plates. Suppose that in the first pointing the image was approaching the
wires ; then bring it into the opposite quarter of the field of view, where the image
recedes from the wires ; bring the wires tangent and read as before. The mean of
these readings will give the apparent altitude and the plate reading for the sun's
centre. If the transit has a full vertical circle, the telescope should be reversed
between the two pointings, to eliminate all errors of adjustment. If the transit has
only a vertical arc, no reversal can be made, and great care must be taken that the
plate levels and standards are in good adjustment and the index error accurately
determined.
" 2. Read plates when pointing at any convenient mark, thus finding the angle
between the sun's centre and the mark.
'* 3. Computation of the PZS triangle enables us to find the angle between the
sun and the north ; then, addition or subtraction of the angle between the sun and
the mark (as the mark is north or south of the sun) gives the angle made by the line
(station to mark), and the meridian.
" The accompanying form of notes and reductions needs but a few remarks :
OBSERVATION.
Telescope.
Horizontal Circlk Readings.
Vertical Circle
Readings.
On Mark.
On Sun.
Date and Time.
Direct^
Reversed -fe.
240" 41' 00"
60" 41' oo"
282° 51' 30"
102" 43' 00"
30" 57' 00'
31° 09' go'
Sept. 26, 1896, 2.30 P.M.,
standard time, 75th merid.
Averages. .
240° 41' 00"
282^ 47' 15"
31* 03' 00'
COMPUTATION.
Declination at Greenwich noon = 7 a.m. standard time 75th
meridian = i" 32' 33" south
Hourly change = 58,5". Change for 7^ hours = 58.5 x 7^^ . = 7' 39" south
Declination at 2 r.M = i" 40' 12" south
* Engineering News, May 20, 1897. Added to the thirteenth edition, 1897.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, lO^a
Average vertical angle by observation = 31** 03' 00"
Correction for refraction . . . = i' 40"
True altitude . . . . = 31° 01' 20"
Latitude of Thayer School = 43° 42' 10"
Station about i mile south = i' 00"
Latitude of station = 43* 41' 10"
Cos I PZS = 4 /sin i ^ X sin {\ 5^ co-decl.)
^ sin co-alt. x sin co-lat.
where S = co-decl. + co-alt. + co-lat.
co-decl. = 91** 40' 12"
co-alt. = 58** 58' 40"
co-lat. = 46* 18' 50"
S = 196" 57' 42"
i5= 98^ 28' 51"
co-decl. = 91" 40' 12"
S — co-decl. = 6° 48' 39"
log sin 98' 28' 51" = 9.995225
•' *• 6" 48' 39" = 9.074052
a. c. '* ** 58° 58' 40" = 0.067035
a. c. ** ** 46* 18' 50" = 0.140781
19.277093
log cos \ PZS = 9.638546
i PZS = 64*" 12' 40"
PZSr=i2%^ 25' 20"
Azimuth of sun from north = 128" 25' 20"
Angle between sun and mark = 42** 06' 15"
Angle north — station — mark = 170° 31' 35"
*• If a single observation is made, the altitude must be changed by the semi-
diameter (16') and the horizontal angle by, not 16', but z — 7—. The dihedral
cos 01 alt.
angle, whos« edge is the vertical line through the instrument subtended by the
semi-diameter» varies with the altitude of the sun, from 16' for alt. = 0°, to 46' for
alt. = 67* on June 21st, in this latitude (43° 42')."
If a colored shield is used over the objective, it should be a
plate glass with parallel surfaces. If the shield is over the eye
end no such parallelism is necessary. For direct observationson
the sun by the eye, a refracting prism at the eye end is necessary.
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I04 SURVEYING.
THE GRADFENTER ATTACHMENT.
104. The Gradienter is a tangent-screw with a micrometer-
head attached to the horizontal axis of the telescope for the
purpose of turning off vertical angles that are expressed in
terms of its tangent as so many feet to the hundred. Such a
device is shown in Fig. 17. In railroad work, the grade or slope
is expressed in this manner, as 26.4 feet per mile, or as 0.5 foot
per 100 feet. The micrometer-head is graduated so that one
revolution raises or lowers the telescope by i foot or 0.5 foot
in 100 feet. It is divided into 100 or 50 parts, so that each
division on the head is equivalent to o.oi foot in loO feet. This
attachment is found very convenient in railroad work. It is
also of general utility in obtaining approximate distances. On
level ground the distance is read directly, but on sloping
ground the rod is still held vertical, and the distance read is too
great. The true horizontal distance may be found by multiply-
ing the distance read by the factors for horizontal distance
given in table V.* Thus, if one revolution of the screw raises
the line of sight i foot at a distance of 100 feet, and if at a cer-
tain unknown distance one revolution of screw caused the line
to pass over 5.5 feet on the rod, then the distance was 550 feet
if the ground was horizontal. If the rod-readings had a mean
vertical angle of 15®, the horizontal distance was 550 X 93.3 —
513 feet.
CARE OF THE TRANSIT.
105. The Transit should be protected from rain and dust
as much as possible. A silk gossamer water-proof bag should
be carried by the observer to be used for this purpose. If water
gets inside the telescope, remove the eye-piece and let it dry
out. If moisture collects between the two parts of the objec-
* This table is for reduction of stadia measurements, and is explained in
the chapter on Topographical Survryinp, Art. 205.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. IO5
tive remove it, and dry it with a gentle heat over a stove or
lamp, but do not separate the glasses. If dust settles on the wires
it may be blown off by removing both objective and eye-piece
and blowing gently through the tube. Dust should be removed
from the glasses by a camershair brush, which should always
be carried for the purpose. A clean handkerchief may be used
with a gentle pressure to prevent scratching in case the dust is
gritty. Use alcohol for cleansing greasy or badly soiled glasses.
No part exposed to dust should be oiled, as this serves to retain
all the dust that may fall on it. The centres should be cleaned
occasionally with chamois skin, and oiled by a very little pure
watch-oil. In the absence of watch-oil plumbago will be found
to serve. A soft lead-pencil may be scraped and a little rubbed
on the spindles with the finger. The tripod legs should have
no lost motion either at the head or in their iron shoes. If the
legs are split, as in Fig. 17, and fastened by thumb-nuts, these
should be loosened when the instrument is carried and tight-
ened again after setting. They may thus be made very tight and
rigid while the instrument is in use without danger of break-
ing the bolts in closing the legs, which is very liable to result
if the screws are not loosened. For a method of putting in new
cross-wires see chapter on Topographical Surveying, Art. 207.
EXERCISES WITH THE TRANSIT.
Z06. Elstablish three stations forming a triangle. Measure the three hori-
zonul angles and see if their sum is iSo"*.
107. Prolong a line in azimuth and distance by carrying both around an
imaginat-y obstruction, and then check the azimuth by a back-sight and the dis-
tance by measurement. Thus, let A and B be two points establishing a line.
The problem is to establish two other points, C and />, in the continuation of
the line AB^ with an imaginary obstruction to both sight and measurement
between B and C. The distance BC is also to be obtained.
The equilateral triangle will be found most efficient.
X08. Find both the distance to and the height of an inaccessible steeple,
chimney, smokestack, or tree.
Measure a baseline such that its two extremities make with the given object
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I06 SURVEYING.
approximately an isosceles triangle (it is desirable that no angle of the triangle
should be less than 30** nor more than 120**). The top of the object only need
be visible from the two ends of the base. Measure both the horizontal and
vertical angles at the extremities of the base-line subtended by the other two
points of the triangle. Let A and B be the extremities of the base and P the
point whose distance and elevation are required. We then have for horizontal
angles
Sin P\sivi A :: AB : BP\
also sin P\%\Xi B :: AB : AP.
In reading the vertical angles to the base-stations the reading should be
taken on a point as high above the ground (or peg) as the telescope is above the
peg over which it is set. The difference in the elevations of the two pegs is
then obtained. The vertical angle to the point P is taken to the summit, and
height of instrument added in each case to find its elevation above peg. If A
be the lower of the two base-stations and if I a and Ib be the heights of instru-
ment (line of sight) above the peg in the two cases, and if K^, Vb, Vp and
Vp be the vertical angles read to the corresponding points, we may write:
Elevation of B above A = AB tan Vb\
'' P '' A- AP tan Vp,
Also, from the vertical angles taken at B, we have:
Elevation of A below B = AB tan Va\
*• P above B = BP tan Vp.
We now have a check on both the relative elevations and on the distances
AP and BP, Assuming the elevation of A to be zero, we have:
Elevation of P above A = AP iSin Vp=zAB un Vb + BP tan Vp'.
This equality will not result unless the observations were well taken, the
computations accurately made, and the instrument carefully adjusted. The ad-
justments mainly involved here are the plate-bubbles and the vernie^on the
vertical circle. If the points are a considerable distance apart, as over a half-
mile, the elevations obtained by reading the vertical angles are appreciably too
great, on account of the earth's curvature. This may be taken as eight inches
for one mile and proportional to the square of the distance. Or. we may write:
Elevation correction on long sights, in inches,* = — 8 (distance in miles^.
If the distances are all less than about half a mile, no attention need be paid
to this correction in this problem.
♦ For a full discussioo of this subject sec chap. XIV.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 10?
109. Find the height of a tree or house above the ground, on a distant hill,
ff ithout going to the immediate locality.
xzo. Find the horizontal length and bearing of a line joining two visible but
inaccessible objects. Use the magnetic bearing if the true bearing of the base-
line is not known.
zzi. Find the horizontal length and bearing of a line joining two inaccessi-
ble points both of which cannot be seen from any one position.
Let A and B be the inaccessible points. Measure a base CD such that A is
seen from C. and B from D. Auxiliary bases and triangles may be used to
find the lengths of yf Cand BD. Knowing -<4Cand CD and the included angle,
compute AD in bearing and distance. The angle ADB may now be found,
which, with the adjacent sides AD and BD known, enables the side ^^ tobe
found in bearing and distance.
1x2. With the transit badly out of level, or with horizontal axis of the tele-
scope thrown considerably out of the horizontal, measure the horizontal angle
between two objects having very different angular, elevations. Do this with
both telescope normal and telescope reversed, and note the difference in the
values of the angle obtained in the two cases.
113. Select a series of points on uneven ground, enclosing an area, and
occupy them successively with the transit, obtaining the traverse angles. That
is, knowing or assuming the azimuth of the first line, obtain the azimuths of the
other connecting lines, or courses, with reference to this one, returning to the
first point and obtaining the azimuth of the first course as carried around by the
traversed line. This should agree with the original azimuth of this course.
The distances need not be measured for this check.
1x4. Lay out a straight line on uneven ground by the method given in Art.
100, occupying from six to ten stations. Return over the same line and estab-
lish a second series of points, paying no attention to the first series, and then
note the discrepancies on the several stakes. In returning, the two final points
of the first line become the initial points of (he second, this return line being a
prolongation of the line joining these two points. If these deviate ever so
little, therefore, from the true line, the discrepancy will increase towards the
initial point.
Similar exercises to those given for the solar compass may be assigned for
the solar attachment.
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io8
SURVEYING,
THE SEXTANT.
115. The Sextant is the most convenient and accurate
hand-instrument yet devised for measuring angles, whether
horizontal, vertical, or inclined. It is called a sextant because
its limb includes but a 60° arc of the circle. It will measure
angles, however, to 120°. It is held in the hand, measures an
angle by a single observation, and will give very accurate re-
sults even when the observer has a very unstable support, as
• Fig. ai.
on board ship. It is exclusively used in observations at sea,
and is always used in surveying where angles are to be meas-
ured from a boat, as in locating soundings, buoys, etc., as well
as in reconnoissance work, explorations, and preliminary sur-
veys. It has been in use since about 173C4
The accompanying cut shows a common form of this in-
strument as manufactured by Fauth & Co., Washington. The
limb has a /^inch radius, and reads to 10 seconds of arc.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, I09
There is a mirror M (Fig. 22), called the /«rfif;r GlasSy rigidly
attached to the movable arm MAy which carries a vernier
reading on the graduated limb CD, There is another mirror,
/, called the Horizon Glass, rigidly attached to the frame of
the instrument, and a telescope pointing into this mirror, also
rigidly attached. This mirror is silvered on its lower half, but
clear on its upper half. A ray of light coming from H passes
Fig. 32.
through the clear portion of the mirror / on through the tele-
scope to the eye at E. Also, a ray from an object at O strikes
the mirror il/; is reflected to m, and then through the telescope
to E. Through one half of the objective come the rays from
H, and through the'other half the rays from O, each of which
sets of rays forms a perfect image. By moving the arm MA
it is evident these images will appear to move over each other.
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no SURVEYING,
and for one position only will they appear to coincide. The
bringing of the two images into exact coincidence is what the
observation consists in, and however unsteady the motion of
the observer may be, he can occasionally see both images at
once, and so by a series of approximations he may finally put
the arm in its true position for exact superposed images.
The angle subtended by the two objects is then read off on
the limb.
ii6. The Theory of the Sextant rests on the optical
principle that ** if a ray of light suffers two successive reflec-
tions in the same plane by two plane mirrors, the angle be-
tween the first and last directions of the ray is twice the angle
of the mirrors.*'
To prove this, let OM and mE be the first and last posi-
tions of the ray, the latter making with the former produced
the angle E. The angle of the mirrors is the angle A. The
angles of incidence and reflection at the two mirrors are the
angles / and /', /W, and/w being the normals.
We may now write :
Angle E = OMm - MmE.
angle A = ImM— mMA
= (9o°-/0-(9O^-0
Therefore E = 2A. Q. E. D.
When the mirrors are brought into parallel planes, the
angle A becomes zero, whence E also is zero, or the rays OM
and Hm are parallel. This gives the position of the arm for
the zero-reading of the vernier. The limb is graduated from
this point towards the left in such a way that a 60° arc of the
circle will read to 120°. That is, a movement of 1° on the arc
really measures an angle of 2® in the incident rays, so it must
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, III
be graduated as two degrees instead of one. The very large
radius enables this to be done without difficulty.
ADJUSTMENTS OF THE SEXTANT.
X17. To make the Index Glass perpendicular to the
Plane of the Sextant.— Bring the vernier to read about 30°
and examine the arc and its image in the index glass to see if
they form a continuous curve. If the glass is not perpendi-
cular to the plane of the arc, the image will appear above or
below the arc, according as the mirror leans forward or back-
ward. It is adjusted by slips of thin paper under the project-
ing points and corners of the frame.
118. To make the Horizon Glass Parallel to the Index
Glass for a Zero-reading of the Vernier.— Set the vernier
to read zero and see if the direct and reflected images of a
well-defined distant object, as a star, come into exact coinci-
dence. If not, adjust the horizon glass until they do. If this
adjustment cannot be made, bring the objects into coincidence,
or even with each other so far as the motion of the arm is con-
cerned, and read the vernier. This is the index error of the
instrument and is to be applied to all angles read. The better
class of instruments all allow the horizon glass to be adjusted.
This adjustment is generally given as two, but it is best con-
sidered as one. If made parallel to the index glass after that
has been adjusted, it must be perpendi<5ular to the plane of
the instrument.
119. To make Jthe Line of Sight of the Telescope
parallel to the Plane of the Sextant— The reticule in the
sextant carries four wires forming a square in the centre of
the field. The centre of this square is in the line of collima-
tion of the instrument.
Rest the sextant on a plane surface, pointing the telescope
upon a well-defined point some twenty feet distant. Place two
objects of equal height upon the extremities of the limb that
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112 SURVEYING,
will serve to establish a line of sight parallel to the limb. Two
lead-pencils of same diameter will serve, but they had best be
of such height as to make this line of sight even with that of
the telescope. If both lines of sight come upon the same
point to within a half-inch or so at a distance of 20 feet,
the resulting maximum error in the measurement of an angle
will be only about i''.
THE USE OF THE SEXTANT.
120. To measure an Angle with the sextant, bring its
plane into the plane of the two objects. Turn the direct line
of sight upon the fainter object, which may require the instru-
ment to be held face downwards, and bring the two images
into coincidence. The reading of the limb is the angle re-
quired. It must be remembered that the angles measured by
the sextant are the true angles subtended by the two objects at
the point of observation^ and not the vertical or horizontal
projection of these angles, as is the case with the transit. The
true vertex of the measured angle is at E, Fig. 21. It is evident
the position of E is dependent on the size of the angle, being
at a^ great distance back of the instrument for a very small
angle. The instrument should therefore not be used for meas-
uring very small angles except as between objects a very great
distance off. The sextant is seldom or never used for measur-
ing angles where the position of the instrument (or the vertex
of the angle) needs to be known with great accuracy.
EXERCISES FOR THE SEXTANT.
Z2I. Measure the altitude of the sun or a star at its culmination by bringing
the direct image, reflected from the surface of mercury held in a flat dish on
the ground, into coincidence with the image reflected from the index glass.
Half the observed angle is the altitude of the body. The altitude of a terres-
trial object may be obtained in the same manner, in which case the vessel of
mercury should rest on an elevated stand ; the sextant could then be brought
near to it and the angular divergence of the two incident rays to the mercury
surface and index glass reduced to an inappreciable quantity.
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ADJUSTMEiWT, USE, AND CARE OF INSTRUMENTS, II3
If the observation of a heavenly body be made on the meridian and the
declination of the body at the time of observation be known, the latitude of the
place is readily found.
122. Measure the angle subtended by two moving bodies, as of two men
walking the street in the same direction, or of two boats on the water, ('ibis is
to illustrate the capacity of the sextant, tor none but a reflecting instrument
bringing ti^ ) converging lines of sight into coincidence is competent to do
this.)
The exercises given in Arts. 106, 108, 109, and no for the transit may also
ser\*e for the sextant. Further applications of the sextant in locating soundings
are given in chap. X.
122a. The Cross-section Polar Protractor. — The accom-
panying cuts (Figs. 24 and 25) illustrate an instrument recently
invented and used on the New York Aqueduct for taking polar
coordinates of the cross-sections of the tunnel. It consists of a
plain circular disk, graduated to single degrees, and mounted
on a tripod in such a way that it may be levelled up and also
have a vertical motion and a motion about the vertical axis.*
The construction is shown clearly in the figures.
In use it is mounted with its centre in the axis of the tun-
nel. A light wooden measuring rod, not shown in the figures,
tapering to a point and shod with brass of sufficient length,
and graduated to feet and hundredths, lies upon the wooden
arm or rest, which revolves upon the face of the disk, and
slides out to a contact with the surface at such points as are
to be taken. If the only information desired is whether
or not the excavation is sufficient, or beyond the established
lines, then the rod is set to the proper radius, and if it swings
clear, the fact is determined. If a true copy of the actual
cross-section is desired, then the rod is brought into contact
with the significant points in the cross-section (mostly the
* Those used on the New York Aqueduct were designed by F. W. Watkins
and Alfred Craven, and were manufactured by Heller and Brightly, Philadelphia.
See description in ** Trans. Am. Soc. Civ. Eng'rs," 1890, and Engineering News,
July 26, 1890.
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114
SURVEYWG,
points of greatest projection and depression of the surface),
and the angle and distance read and recorded. In the instru-
ment here shown, the graduation increases in both directions
from the top to i8o° at the bottom. Perhaps a better arrange-
ment would be to have the angles increase continuously to
f"
0_
\\ -1
\
A
i„
Fig. 24.
Fig.
360°. The work could be plotted by means of such a pra-
tractor as shown in Figs. 64 or 66, Chapter VIII. The points
being plotted, they should be joined by a free-hand line and
the area determined by the planimeter.
If the cross-section contains one or more marks from which
the axis of the tunnel may be found, as an alignment mark
and a bench mark (which may be one and the same), then the
instrument may be set up at random on this section, and these
fixed marks pointed in and plotted, along with the cross-sec-
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. II5
tion points. The axis of the tunnel can then be laid off from
the plotted marks, and by drawing in the established lines
from this axial point, the question of clearance may be deter-
mined nearly as well as by setting the instrument in the axis
of the tunnel itself. The actual cross-section and area are
quite as well determined as if the instrument were carefully
centered on the axial line.
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lib SURVEYING.
CHAPTER V.
THE PLANE TABLE.
123. The Plane Table consists of a drawing-board
properly mounted on which rests an aHdade carrying a line of
sight rigidly attached to a plain ruler with a fiducial edge.
The line of sight is usually determined by a telescope, as in
Fig. 26. This telescope has no lateral motion with respect to
the ruler, but both may be moved at pleasure on the table.
The telescope has a vertical motion on a transverse axis, as in
the transit. It is also provided with a level tube, either
detachable or permanently fixed. The table is levelled by
means of one round or two cross bubbles on the ruler of the.
alidade. The line of sight of the telescope is usually paralle)
to the fiducial edge of the ruler, though this is not essential.
It is only necessary that they should make a fixed horizontal
angle with each other. The table itself must have a free hori-
zontal angular movement and the ordinary clamp and slow-
motion screw. The table corresponds to the graduated limb
in the transit, the alidades in the two instruments performing
similar duties. Instead, however, of reading off certain hori-
zontal angles, as is done with the transit, and afterwards
plotting them on paper, the directions of the various pointings
are at once drawn on the paper which is mounted on the top
of the table, no angles being read. The true relative positions
of certain points in the landscape are thus transferred directly
to the drawing-paper to any desired scale. The magnetic
bearing of any line may be determined by means of the declu
natoTy which is a small box carrying a needle which can swing
some ten degrees either side of the zero-line. The zero-line
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ADJUSTMENT, USE, AND CARE OF mSTRUMENTS. H)
i
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Il8 SURVEYING.
being parallel to one edge of the box, the magnetic meridian
may be at once marked down on any portion of the map, and
the bearing of any intersecting line determined by means of a
protractor. The instrument has been long and extensively used
for mapping purposes, and is still the only instrument used
for the ** filling-in" of the topographical charts of the U. S.
Coast and Geodetic Survey. An extended account of the
instrument and the field methods in use on that service may
be found in Appendix 13 of the Report of the U. S. Coast and
Geodetic Survey for 1880. The following discussion is partly
from that source.
ADJUSTMENTS OF THE ALIDADE.
124. To make the Axes of the Plate-bubbles parallel
to the Plane of the Table. — Level the table with the alidade
in any position, noting the readings of the bubbles. Mark the
exact position of the alidade on the table, take it up carefully,
and, reversing it end for end, replace it by the same marks. If
the bubbles now have the same readings as before, with refer-
ence to the table they are parallel to the plane of the table.
If not, adjust the bubbles for one half the movement and try
again.
125. To cause the Line of Sight to revolve in a Vertical
Plane. — This adjustment is the same as in the transit. It need
not be made with such extreme accuracy, however, and the
plumb-line test is sufficient. With the instrument carefully
levelled, cause the line of sight to follow a plumb-line through
as great an arc as convenient. If the line of sight deviates
from the plumb-line raise or lower one end of the transverse
axis of the telescope, until it will follow it with sufficient exact-
ness.
126. To cause the Telescope-bubble and the Vernier on
the Vertical Arc to read Zero when the Line of Sight is
Horizontal. — This adjustment is also the same as in the
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, I I9
transit. The methods given for the transit may be used with
the plane table, or a sea horizon may be used as establishing a
horizontal line, or a levelling-instrument may be set up beside
the plane table having the telescopes at the same elevation, and
both lines of sight turned upon the same point in the horizontal
plane as determined by the level. The bubble and vernier are
then both adjusted to this position of telescope.
This adjustment is important if elevations are to be deter-
mined either by vertical angles or by horizontal lines of sight.
If only geographical position is sought this adjustment may
be neglected.
THE USE OF THE PLANE TABLE.
127. In using the Plane Table at least two points on the
ground, over which the table may be set, must be plotted on
the paper to the scale of the map before the work of locating
other points can begin. This requires that the distance between
these points shall be known, which distance becomes the base-
line for all locations on that sheet. Any error in the measure-
ment or plotting of this line produces a like proportional error
in all other lines on the map.
The plane table is set over one of these plotted points, the
fiducial edge of the ruler brought into coincidence with the two
points, and the table revolved until the line of sight comes on
he distant point. The table is now clamped and carefully set
by the slow-motion screw in this position, when it is said to be
oriented^ or in position.
In Figs. 27 and 28, let T, V T," T,'" represent the plane-
table sheet and the points a and/ the original plotted points.
The corresponding points on the ground are A and P, the latter
being covered by/ in Fig. 27, and the former hy a in Fig. 28.
In Fig. 27, the plotted point / is centred over the point P, the
ruler made to coincide with /?/, and the telescope made to read
on A by shifting the table. For plotting the directions of
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/
/
/
t'
1
f
/
hYJp
-^^ - - 1
,1
y
t"
Fig. »7.
.i<
Pig. 28.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 121
other objects on the ground, the alidade is made to revolve about
p just as the transit revolves about its centre. A needle is
sometimes stuck at this point, and the ruler caused to press
against it in all pointings, but this defaces the sheet. Other
pointings are now made to By C, and D, which may be used as
stations, and also to a chimney (^//.). a tree (/.), a cupola
{cup,\ a spire (sp.)^ and a windmill {w.m.). Short lines are
drawn at the estimated distance from p, and these marked with
letters, as in the figure, or by numbers, and a key to the numbers
kept in the sketch- or note-book.
The table is now removed to A^ the other known point, and
set with the point a on the plot over the point A on the
ground, when the table is approximately oriented. The ruler
is now set as shown in Fig. 28, coinciding with a and/, but
pointing towards/. The table is then swung in azimuth until
the line of sight falls on P, when it is clamped. It is now
oriented * for this station, and pointings are taken on all the
objects sighted from P, and on such others as may be sighted
from subsequent stations, the alidade now revolving about the
point a on the paper. The intersections on the plot of the
two pointings taken to the same object from A and P will evi-
dently be the true position on the plot for those points with
reference to, and to the scale of, the line ap. These intersec-
tions are shown in Fig. 28.
It is evident that if other points, as D or C, be now occu-
pied, the table oriented on either A or P, and pointings taken
on any of the objects sighted from both A and P, the third or
fourth line drawn to the several objects should intersect the
first two in a common point. This furnishes a check on the
work, and should be taken for all important points. It is pref-
erable also to have more than two points on the sheet pre-
♦ It will be noted that this process of orienting the plane table is practically
identical with that by which the limb of the transit is oriented in traversing
(arL loi).
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122 SURVEYTNG.
viously determined. Thus, if B were also known and plotted
at b, when the table had been oriented on any other station,
and a pointing taken to B, the fiducial edge of the ruler
should have passed through b.
As fast as intersections are obtained and points located
the accompanying details should be drawn in on the map to
the proper scale. If distances are read by means of stadia
wires on a rod held at the various points (see chap. VIII),
then a single pointing may locate an object, the distance being
taken off from a scale of equal parts, and the point at once
plotted on the proper direction-line. It is now common to do
this in all plane-table surveying.
128. Location by Resection. — This consists in locating
the points occupied by pointings to known and plotted points.
The simple case is where a single pointing has been taken to
this point from some known point, and a line drawn through
it on the sheet. It is not known what point on this line
represents the plotted position of this station. The setting of
the instrument can therefore be but approximate, but near
enough for all purposes. The table can be oriented as before,
there being one pointing and corresponding line from a known
point. A station is then selected, a pointing to which is as
nearly 90 degrees from the orienting line as possible, and the
alidade so placed that while the telescope sights the object the
fiducial edge of the ruler passes through the plot of the same
on the sheet. The intersection of this edge with the former
line to this station gives the station's true position on the
sheet. This latter operation is called resection. Another re-
section from any other determined point may be made for a
check.
129. To find the Position of an Unknown Point by Re-
section on Three Known Points. — This is known as the
Three-point Problem, and occurs also in the use of the sextant
in locating soundings. It is fully discussed in that connection
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, I23
(see Art. 228), so that only a mechanical solution suitable
for the problem in hand will be given here. It is under-
stood there are three known points,-^, 5, and C, plotted in
a, b, and c on the map. The table is set up over anv s;*»ven
point (not in the circumference of a circle through A, B, and
C). Fasten a piece of tracing-paper, or linen, on the board,
and mark on it a point / for the station P occupied. Level
the table, but of course it cannot be oriented. Take pointings!
to A, By and C, and draw lines on the tracing-paper from /
towards ^, 6, and Cy long enough to cover these distances when
drawn to scale. Remove the alidade and shift the tracing-
paper until the three lines drawn may be made to coincide
exactly with the three plotted points a, by and c. The point
p is then the true position of this point on the sheet. This
being pricked through, the table may now be oriented and the
work proceed as usual.
130. To find the Position of an Unknown Point by Re-
section on Two Known Points. — This is called the Two-
point Problem, and but one of several solutions will be given.
It is evident that if the table could be properly oriented over
the required point, its position on the sheet could be at once
found by resection on the two known points. The table may
be oriented in the following manner: Let A and B be the
known points plotted in a and b on the sheet. Let C be the
unknown point whose position c on the sheet is desired.
Select a fourth point Dy which may be occupied, and so placed
that intersections from C and D ox\ A and B will give angles
between 30 and 120 degrees. Fasten a piece of tracing linen
or paper on the board, marking a point d' at random. Set
up over Dy orienting the table as nearly as may be by the
needle or otherwise. Draw lines from d' towards Ay B, and
C. Mark ofif on the latter the estimated distance to C, to
scale, calling this point C. Set up over C, with c' over the
station, orienting on D by the line (/d\ This brings the table
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124 SURVEYING,
parallel to its former position at D, From c' draw lines to A
and J9, intersecting the corresponding lines drawn from d in
a' and b'. We now have a quadrilateral a'b'c'd! similar to
the quadrilateral formed by the true positions of the plotted
points abed, but it differs in size, since the distance c'd! was
assumed, and also in position (azimuth), since the table was
not properly oriented at either station. Remove the alidade,
and shift the tracing until the line a'b' coincides with a and b
on the sheet. Replace the alidade on the tracing, bringing it
into coincidence with c'a\ c'b\ or c'd\ and revolve the table on
its axis until the line of sight comes upon^, B, or D^ as the
case may be. The table is now oriented, when the true posi-
tion of c may be readily found by resecting from a and b^
which, when pricked through, gives its position on the sheet.
The student may show how the same result could have been obtained with-
out the aid of tracing-paper.
If the fourth point D may be taken in range "with A and B^
the table may be properly oriented on this range, and a line
drawn towards C from any point on this range line on the plot.
Then C is occupied, and the table again properly oriented by
this line just drawn, when the true position of c may be found
by resecting from a and b, as before.
In general, if the table can be properly oriented over any
unknown point from which sights may be taken to two or
more known (plotted) points, the position of this unknown
point is at once found by resection from the known points.
The student would do well to look upon the table and the
attached plot as analogous to the graduated horizontal limb in
the transit. The principles and methods of orienting are pre-
cisely similar, the pointings differing only in this, that with the
transit the horizontal angle, referred to the meridian, is read
off, recorded, and afterwards plotted, while with the plane
table this bearing is immediately drawn upon the sheet.
131. Tlie Measurement of Distances by Stadia. — This
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 12$
method of determining short distances is now generally used in
connection with the plane table. It is fully discussed in chap-
ter VIII., where the principles ot its action and its use with
the transit are given at length. The same principles, field
methods, and tables apply to its use with the plane table,
with such modifications as one accustomed to the use of the
plane table would readily introduce. When used in this way
it enables a point to be plotted from a single pointing, it
being located by polar coordinates (azimuth and distance), in-
stead of by intersections.
EXERCISES WITH THE PLANE TABLE.
133. Make a plane table survey of the college campus, measuring the length
of one side for a base.
133. Having located several points on the sheet by intersections y occupy
them and check their location by resection.
134. Locate a point (not plotted) by resection on three known points (art.
129).
135. Locate a point (not plotted) by resection on two known points, first
taking the auxiliary point D not in line with AB, and then by taking it in line
with AB. This gives a check on the position of the point, and shows the ad-
vantages of the second method when it is feasible.
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126 SURVEYING.
CHAPTER VI.
ADDITIONAL INSTRUMENTS USED IN SURVEYING AND
PLOTTING.
THE ANEROID BAROMETER.
136. The Aneroid Barometer consists of a circular me-
tallic box, hermetically sealed, one side being covered by a
corrugated plate. The air is mostly removed, enough only
being left in to compensate the diminished stiffness of the cor-
FlG. 39.
rugated cover at higher temperatures. This cover rises or
falls as the outer pressure is less or greater, and this slight
motion is greatly multiplied and transmitted to an index
pointer moving over a scale on the outer face. The motion
of the index is compared with a standard mercurial barom-
eter and the scale graduated accordingly. Inasmuch as all
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 12^
barometric tables are prepared for mercurial barometers,
wherein the atmospheric pressure is recorded in inches of
mercury, the aneroid barometer is graduated so that its read-
ings are identical with those of the mercurial column.
Figure 29 shows a form of the aneroid designed for eleva-
tions to 4000 feet above or to 2000 feet below sea-level. It
has a vernier attachment and is read with a magnifying-glass
to single feet of elevation. It must not be supposed, how-
ever, that elevations can be determined with anything like this
degree of accuracy by any kind of barometer. The barometer
simply indicates the pressure at the given time and place, but
for the same place the pressure varies greatly from various
causes. All barometric changes, therefore, cannot be attrib-
uted to a change in elevation, when the barometer is carried
about from place to place.
If two barometers are used simultaneously, which have
been duly compared with each other, one at a fixed point of
known elevation and the other carried about from point to
point in the same locality, as on a reconnoissance, then the
two sets of readings will give very close approximations to
the differences of elevation. If the difference of elevation be-
tween distant points is desired, then long series of readings
should be taken to eliminate local changes of pressure. The
aneroid barometer is better adapted to surveys than the mer-
curial, since it may be transported and handled with greater
ease and less danger. It is not so absolute a test of pressure,
however, and is only used by exploring or reconnoissance ^
parties. For fixed stations the mercurial barometer is to be
preferred. It has been found from experience that the small
aneroids of if to 2j inches diameter give as accurate results
as the larger ones.
137. Barometric Formute.— In the following derivation
of the fundamental barometric formula the calculus is used, so
that the student will have to take portions of it on trust until
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128 SURVEYING,
he has studied that branch of mathematics. All that follows
Eq. (4) he can read.
Let H =^ height of the ** homogeneous atmosphere'** in lat.
45°.
h = corresponding height of the mercurial column.
d = the relative density of the ** homogeneous atmos-
phere" with reference to mercury.
z= difference of elevation between two points, with
barometric readings of A' and A„ at the higher
and at the lower point respectively.
Then from the equilibrium between the pressures of the
mercurial column and atmosphere we have :
A = dH (I)
Also, for a small change in elevation, ds, the corresponding
change in the height of the mercurial column would be
dA = dds (2)
Substituting in (2) the value of d as given by (i), we have :
dA = -jydz',
or, ds:=^H-j (3)
i^ntegrating (3) between the limits //' and A, we have:
*=.^i^*x = ^i°g4' • • • • (4)
* ** Homogeneous atmosphere'* signifies a purely imaginary condition
wherein the atmosphere is supposed to be of uniform density from sea-level to
such upper limit as may be necessary to give the observed pressure at the ob-
served temperatur««
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 1 29
where the logarithm is in the Napierian system. Dividing by
the modulus of the common system to adapt it to computation
by the ordinary tables, we have :
z^2.i02i%H\oz^ -^ (5)
If Ho be the height of the homogeneous atmosphere at a
temperature of 32° F., and if h^ be the height of the mercurial
column at sea-level at same temperature, and if g^ and g^ be
the specific gravities of mercury and air respectively, then,
evidently,
/f„=tet (6)
From experiment we have :
ho = 29.92 inches,
^m= 13.596
ga = 0.001293
whence H^ = 26,220 feet.
This is on the assumption that gravity is constant to this
height above sea-level. When this is corrected for variable
gravity we have :
Ho = 26,284 feet (7)
Equation (7) gives the height of the homogeneous atmos-
phere at a temperature of 32° F. But since the volume of a gas
under constant pressure varies directly as the temperature, and
since the coefficient of expansion of air is 0.002034 for i** F.,
we have for the height of the homogeneous atmosphere at any
temperature :
/f=//'o [I +0.002034 (/-- 32°)] ... (8)
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I30 SURVEYING,
If the temperature chosen be the mean of the temperatures at
the two points of observation, as /' and /, for the upper and
lower points respectively, then we should have :
H^ H, [i + 0.002034 (^--32)]
= 26,284 [i + 0.001017 (/'+ /,— 64)] . . (9)
Substituting this value of H in Eq. (5) we obtain :
h
z = 60,520 [l + O.OOIOI7 (/'+ /, — 64)] log -rf. . (10)
If we wish to refer this equation to approximate sea-level
(height of mercurial column of 30 inches) and to a mean tem-
perature of the two stations of 50° F., we may write :
30
1 *i , A' 1 30 , 30
Also, when ;'-{-;,= \oo°, we have
/' + ;.- 64 =36°.
Substituting these equivalents in eq. (lo), we obtain
s = 60520 (i + 0.001017X 36) (log ~ — log ^),
« = 62737 log ^ - 62737 log J 00
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I3I
In this equation, the two terms of the second member rep-
resent the elevations of the upper and lower points respec-
tively, above a plane corresponding to a barometric pressure of
30 inches and for a mean temperature of the two positions of
50° F.
Table I. is computed from this equation, the arguments be-
ing the readings of the barometer, h' and A, at the upper and
lower stations respectively, the tabular results being elevations
above an approximate sea-level. The difference between the
two tabular results gives the difference of elevation of the two
points, for a mean temperature of 50° and no allowance made
for the amount of aqueous vapor in the air. For other tem-
peratures, and for the effect of the humidity (which is not ob-
served, but the average conditions assumed to exist), a certain
correction needs to be applied, which correction is not an abso-
lute amount, but is always a certain proportion ol th^ Ax&tx^nc^
of elevation as obtained from eq. (11) or table I. If the two
elevations taken from the table be called A' and A^, and the
correction for temperature and humidity be C, we would have
z=^{A'-A,){i + C) (12)
It is seen, therefore, that C is a coefficient which, when mul-
tiplied into the result obtained from table I., gives the correc-
tion to be applied to that result. The values of C are given
in table II. for various values of /' + A-
The following example will illustrate the use of the tables :
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132
SURVEYING,
TABLE I.
BAROMETRIC ELEVATION.*
Containing A — (
52737 log
\ . Argunient, h,
h
A.
A,
Dif. for
.01.
k.
A.
Dif. for
• CI.
1 k.
1
A.
Dif. for
.01.
Inches
Feet.
Feet.
I Inches.
Feet.
Feet.
. Inches.
Feet.
Feet.
II. O
27.336
—24.6
14.0
20,765
-195
! 17.0
15.476
— 16.0
II. I
27,090
24.4
14. 1
20,570
193
17.1
15.316
15.9
II. 2
20,846
24.2
14.2
20,377
19. 1
17.2
1
15.157
15-8
II. 3
26,604
24.0
14.3
20,186
18.9
1 17.3
14.999
15-7
II. 4
26,364
23.8
14.4
19.997
18.8
17.4
14,842
15.6
II. 5
26,126
23.6
14.5
19,809
18.6
' 17.5
14,686
15.5
II.6
25.890
23.4
14.6
19.623
18.6
1 17.6
14.531
15.4
II. 7
25,656
23.2
14.7
19.437
18.5
, 17.7
14.377
15.4
II. 8
25.424
23.0
14.8
19.252
18.4
1 17.8
14.223
15.3
II.9
25.194
22.8
14.9
19,068
18.2
; 17.9
14,070
15.2
12. 0
24,966
22.6
15.0
18.886
18. 1
18.0
13.918
15. 1
12. 1
24.740
22.4
15. 1
18,705
18.0
' 18. 1
13.767
15.0
12.2
24,516
22.2
15.2
18.525
17.9
. 18.2
13.617
14.9
12.3
24,294
22.1
15.3
18.346
17.8
18.3
13.468
14.9
12.4
24,073
21.9
15.4
18,168
17.6
1 18.4
13.319
14.7
12.5
23.854
21.7
15.5
17.992
175
, 18.5
13.172
14.7
12.6
23,637
21.6
15.6
17.817
17.4
' T8.6
13.025
14.6
12.7
23,421
21.4
1 15.7
17.643
17.3
1 18.7
12,879
14.6
12.8
23,207
21.2
15.8
17.470
17.2
■ 18.8
12.733
14.4
12.9
22,995
21.0
15.9
17,298
17. 1
18.9
12,589
14.4
13.0
22.785
20.9
16.0
17,127
16.9
19.0
12.445
14.3
13. 1
22,576
20.8
I 16. 1
16,958
16.9
I'Q.i
12,302
14.2
13.2
22,368
20.6
1 16.2
16,789
16.8
19.2
12,160
14.2
13.3
22,162
20.4
16.3
16,621
16.7
' 19-3
12,018
14. 1
13.4
21,958
20.1
1 16.4
1
16.454
16.6
19.4
11,877
14.0
13.5
21.757
20.0
I 16.5
16.288
16.4
! 19.5
11.737
13-9
136
21,557
19.9
' 16.6
1
16,124
16.3
19.6
11,598
13.9
13.7
21,358
19 8
16.7
15 961
16.3
19.7
11,459
13.8
13.8
21,160
19.8
16.8
15.798
16.2
1 19.8
11,321
13.7
13.9
20,962
-19.7
16.9
15.636
— 16.0
19.9
11,184
-13.7
14.0
20.765
17.0
T5-476
1
, 20.0
11.047
*This table taken from Appendix 10, Report U. S. Coast and Geodetic
SurvcY, 188 1. / ^f^niXo
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ADJUSTMEXT, USE, AND CARE OF INSTRUMENTS. I33
TABLE I. Barometric Elevation. — Continued,
30
Containing A — 62737 Jog ^ • Argument, h,
n
k.
A.
Inches
Fett.
20.0
11,047
20.1
10,911
20.2
10.776
20.3
10,642
20.4
10,508
20.5
10.375
20.6
10,242
20.7
10,110
20.8
9.979
20.9
9.848
21.0
9.718
21. 1
9.589
21.2
9.460
21.3
9332
21.4
9.204
21.5
9.077
21.6
8.951
21.7
8.825
21.8
8,700
21.9
8.575
22.0
8.451
22.1
8.327
22.2
8,204
22.3
8.082
22.4
7,960
22.5
7.838
22.6
7.717
22.7
7.597
22.8
7.477
22.9
7.358
23 0
7.239 1
Dif. for
.01.
Feet.
— 13.6
13.5
13.4
13.4
13.3
13.3
13.2
13. 1
13. 1
13-0
12.9
12.9
12.8
12.8
12.7
12.6
12.6
12.5
12.5
12.4
12.4
12.3
12.2
12.2
12.2
^12.1
12.0
12.0
II. 9
-II.9
h.
Inches.
23.0
23.1
23.2
23.3
23.4
23.5
23.6
23.7
23.8
23.9
24.0
24.1
24.2
243
24.4
24.5
24.6
24- 7
24.8
24.9
25.0
25.1
25.2
253
254
25.5
25.6
25-7
25.8
25.9
26.0
A.
Feet.
7.239
7,121
7,004
6.887
6.770
6,654
6,538
6,423
6,308
6,194
6,080
5.967
5,854
5.741
5,629
5.518
5.407
5.296
5.186
5.077
4,968
4.859
4.751
4.643
4.535
4.428
4.321
4.215
4,109
4.004
3.899
Dif. for
.ot.
Feet.
-II. 8
II. 7
II. 7
II. 7
II. 6
II. 6
II. 5
II. 5
II. 4
II. 4
II. 3
II. 3
II. 3
II. 2
II. I
II. I
II. I
II. o
10.9
10.9
10.9
10.8
10.8
10.8
10.7
10.7
10.6
10.6
10.5
-10.5
h.
Inches.
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.8
27.9
28.0
28.1
28.2
28.3
28.4
28.5
28.6
28.7
28.8
28.9
29.0
Feet.
3.899
3.794
3,690
3.586
3.483
3,380
3.277
3.175
3.073
2,972
2,871
2.770
2,670
2,570
2.470
2,371
2,272
2,173
2,075
1.977
1,880
1.783
1,686
1,589
1.493
1.397
1,302
1,207
1,112
1,018
924
Dif. for
Feet.
-10.5
10.4
10.4
10.3
10.3
10.3
10.2
10.2
10. 1
10. 1
10. 1
10. o
10. o
10. o
9.9
9-9
9.9
9.8
9.8
9.7
9.7
9.7
9-7
9.6
9.6
9.5
9-5
9.5
9-4
-9.4
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134
SURVEYING.
TABLE I. Barometric Elevations. — Continued.
30
Containing A = 62737 log 1 • Argument, h.
,h.
A.
Dif. for
.01.
h.
A.
Dif. for
.01.
k.
A.
Dif. for
.01.
Inches.
Feet.
Feet.
Inches.
Feet.
Feet.
Inches.
Feet.
Feet.
29.0
924
-9.4
29.7
274
—9.2
30.4
361
—9.0
29.1
830
9.4
29.8
182
9.1
30.5
451
8.9
29.2
29.3
736
643
9-3
9-3
29.9
30.0
91
00
9.1
9.1
30.6
30.7
540
629
8.9
8.8
29.4
550
9.2
30.1
-91
9.0
30.8
1^1
8.8
29.5
458
9.2
30.2
181
9.0
30.9
805
-8.8
29.6
366
—9.2
30.3
271
-9.0
31.0
-893
29.7
274
30.4
361
TABLE n.
CORRECTION COEFFICIENTS TO BAROMETRIC ELEVATIONS
FOR TEMPERATURE AND HUMIDITY.*
r, + /'.
c.
/i-f/'.
c.
h^-t'.
c.
0°
—0. 1025
60
—0.0380
120
+0.0262
5
- .0970
65
— .0326
125
+ -0315
10
- .0915
70
- .0273
130
+ .0368
15
- .0860
75
— .0220
135
+ .0420
20
- .0806
80
- .0166
140
+ .0472
25
- .0752
85
— .0112
145
+ .0524
30
- .0698
90
— .0058
150
+ .0575
35
- .0645
95
— .0004
155
+ .0626
40
- .0592
100
+ .0049
160
+ .0677
45
- .0539
105
+ .0102
165
+ .0728
50
- .0486
1
+ .0156
170
+ .0779
55
- .0433
"5
+ .0209
175
4- .0829
60
— .0380
120
+ .0262
180
+ -0879
♦This table compiled from tables I. and IV. of Appendix 10 of Report of
the U. S. Coast and Geodetic Survey for 188 1.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, I35
Example.
From observations made at Sacramento, CaL, and at Sum-
irit on the top of the Sierra Nevada Mountains, the annual
means were :
h' = 23.288 in. /' = 42.1 F.
K = 30.014 in. /, = 59.9.
From table I. we have
A' •=• 6901.0 feet.
^, = — 12.7 "
A'-A,:=^ 6913.7 •*
From table II. we find for t' -^t^^ i02°.o, ^7= + .0070.
. • . ^ = 6913.7 (i + .0070) = 6962 feet.
138. Use of the Aneroid.— Tiie aneroid barometer should
be carried in a leather case, and it should not be removed from
it. It should be protected from sudden changes of tempera-
ture, and when observations are made it should have the
temperature of the surrounding outer air It should not be
carried so as to be affected by the heat of the body, and should
be read out of doors, or at least away from all artificially
warmed rooms. Always read it in a horizontal position. The
index should be adjusted by means of a screw at its back, to
agree with a standard mercurial barometer, and then this ad-
justment left untouched.
When but a single instrument is used it is advisable io pass
between stations as rapidly as possible, but to siop^X a number
of stations during the day for a half-hour or so, reading the
barometer on arrival and on leaving. The difference of these
two readings shows the rate of change of barometric readings
due to changing atmospheric conditions, and from these iso-
lated rates of change a continuous correction-curve can be con-
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136
SURVEYING.
structed on profile or cross-section paper from which the
instrumental corrections can be taken for any hour of the
day.* The observations should be repeated the same day in
reverse order, the corrections applied as obtained from this
correction curve, and the means taken. Observations should
be made when the humidity of the air is as nearly constant as
possible, and never in times of changeable or snowy weather.
Let the student measure the heights of buildings, hills, etc.,
and then test his results by level or transit.
To interpolate elevations between two points whose elevations
are known^ take a reading at the first known point and pass
rapidly toward the other known point, taking intermediate
readings at determinate distances, and so continue until the
second point is reached, when a reading is also taken here. The
error in the determination between the two known points may
Dlf, of
Elev,
B
Time of Readings Av, Gorr^d Bar, Readings
Fic. 29a. Pig. 09^.
now be distributed along the line either proportionately to the
differences of elevation or to the intervening time, or to both,
in the following manner. To correct for the time changes in
the barometric readings, return at once to the first known point,
and take duplicate readings at all the intervening points at
which readings were previously taken. Note the discrepancy
between the first and final barometric readings at the initial
station, and correct all intermediate readings for their corres-
ponding time intervals. This can be readily done graphically
by taking time as one co-ordinate and the final discrepancy at
the initial station as the other, and joining the two extreme
points of these lines by a straight line, as indicated in Fig. 2ga,
* Mr. Chas. A. Asbburner, Geologist of the Penn. Gcol. Survey, has used
this method with good results.
Barometric
--^
Discrepancy
^^^
at Sta. A
^.^.^-'''''^
A A...-<^
1
8
%
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ADJUSTMENT, USE, A AD CARE OF INSTRUMENTS. 13/
Having thus made the time correction, the barometric read-
ings may be corrected for elevation by another graphical dia-
gram, such as is shown in Fig. 29^, wherein the co-ordinates are
the average corrected barometric readings at the several stations
and the known difference of elevation of the terminal station.
This known difference of elevation thus gives a true interpreta-
tion of all barometric differences independent of the use of any
particular formula. Presumably the time corrections have
included also temperature changes.
This method of leveling between known points is commonly
employed in Switzerland, and usually by going over the route
but once. In this case the time change could be corrected by
reading a duplicate barometer continuously at some stationary
point in the immediate vicinity.
Fig. 30.— Front View. Fig. 31.-— Back View.
THE PEDOMETER.
139. The Pedometer is a pocket-instrument for register-
ing the number of paces taken when walking. It is generally
made in the form of a watch, the front and back views being
shown in Figs. 30 and 31.
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138
SURVEYING,
When the instrument is attached to the belt in an upright
position, as here shown, the jar given it at each step causes the
weighted lever shown in Fig. 31 to drop upon the adjustable
screw 5. The lever recovers its position by the aid of a spring,
and in so doing turns a ratchet-wheel by an amount propor-
tional to the amplitude of the lever*s motion. This may be
adjusted to any length of pace by means of the screw 5, which
is turned by a key. The face is graduated like that of a watch,
and gives the distance travelled in miles. This instrument
may also be used on a horse, and when adjusted to the length
of a horse's step will give equally good results. ' The accuracy
of the result is in proportion to the uniformity of the steps,
after having been adjusted properly for a given individual.
The instrument is only used on explorations, preliminary sur-
veys, and reconnoissance-work.
The Length of Men's Steps has been investigated by Prof.
Jordan,* of the Hanover Polytechnic School. From 256
step-measurements by as many different individuals, of lines
from 650 to 1000 feet in length, carefully measured by
rods and steel tapes, he concludes that the average length of
step is 2.648. feet, ranging from 2.066 to 3.182 feet. The mean
deviation from this amount for a single measurement was
± 0.147 feet, or 5^ per cent. The average age of the persons
making these step-measurements was 20 years. The length of
step decreases with the age of the individual after the age of
25 to 30 years. It is also proportional to the height of the
person. The results for 18 different-sized persons gave the
following averages :
Height of person
5 '.08
5'.25
5'.4i
5'.58
5'.74
5'.90
6'.o7
6'.23
6'. 40
6'.56
Length of step..
2.46
2.53
2.56
2.59
2.62
2.69
2.72
2.76
2.79
2.85
♦ See translation in Engineering News for July 25, 1885.
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ADJUSTMENT, VSE, AND CAkE OF INSTRUMENTS. I39
On slopes the step is always shorter than on level ground,
whether one goes up or down. The following averages from
the step-measurement of 136 lines on mountain-slopes along
trails were found :
Slooc
0"
5^
10'*
15'
20**
25°
30^
1.25
Length of step in ascending
2'.53
2'.30
2'.03
I '.84
i'.64
I '.48
Length of step in descending
2'.53
2'43
2'.36
•
2. 30
2'.20
i'-97
i'.64
The length of the step is also found to increase with the
length of the foot. One steps farther when fresh than when
tired. The increase in the length of the step is also in nearly
direct proportion to the increase of speed in walking.
When the proper personal constants are determined, and
when walking at a constant rate, distances can be determined
by pedometer, or by counting the paces, to within about two
per cent of the truth. One should always take his natural step^
and not an artificial one which is supposed to have a known
value, as three feet, for instance. Let a base be measured off
and each student determine the length of his natural step when
walking at his usual rate, or, what is the same thing, find how
many paces he makes in 100 feet. He then has always a
ready means of determining distances to an approximation,
which in many kinds of work is abundantly sufficient.
THE ODOMETER.
140. The Odometer is an instrument to be attached to
the wheel of a vehicle to record the number of revolutions
made by it. One form of such an instrument is shown in
Fig. 32 attached to the spokes of a wheel.
Each revolution is recorded by means of the revolution of
an axis with reference to the instrument, this axis really being
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I40
SURVEYING.
held stationary by means of an attached pendulum which does
not revolve. The instrument really revolves about this fixed
axis at each revolution of the wheel, and the number of times
Fig.
it does this is properly recorded and indicated by appropriate
gearing and dials.
This method of measuring distances is more accurate than
by pacing, as the length of the circumference of the wheel is a
constant. This length multiplied by the number of revolu-
tions is the distance travelled. It is mostly used by exploring
parties and in military movements in new countries which have
not been surveyed and mapped.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I4T
THE CLINOMETER.
141. The Clinometer is a hand-instrument for determin.
ing the slope of ground or the angle it makes with the horizon.
It consists essentially of a level bubble, a graduated arc, and a
line of sight, so joined that when the line of sight is at any angle
to the horizon the bubble may be brought to a central position
and the slope read off on the graduated arc. Such a combina-
tion is shown in Fig. 33. It is called the Abney level and
Fig. 33.
clinometer, being really a hand-level when the vernier is set to
read zero. The position of the bubble is visible when looking
through the telescope, the same as with the Locke hand-level,
shown in Fig. 16, p. 82. The body of the tube is made square,
so that it may be used to find vertical angles of any surface by
placing the tube upon it and bringing the bubble to the centre.
The graduations on the inner edge of the limb give the slope
in terms of the relative horizontal and vertical components of
any portion of the line ; thus, a slope of 2 to i signifies that
the horizontal component is twice the vertical. In reading this
scale the edge of the vernier-arm is brought into coincidence
with the graduation.
This instrument is very useful in giving approximate slopes
in preliminary surveys, the instrument being pointed to a posi-
10
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142
SURVEYING,
tion as high above the ground as its own elevation when held
to the eye.
THE OPTICAL SQUARE.
142. The Optical Square is a small hand-instrument used
to set off a right angle. It is shown in Fig. 34, the method of
Its use being evident from the figure. Thus, while the rod at
0 is seen directly through the opening, the rod at / is seen in
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 143
the glass as the prolongation downwards of that of o, it being
reflected from the mirrors / and c in succession, they having
an angle of 45** with each other. By this means a line may be
located at right angles to a given line at a given point, or a
point in a given line may be found in the perpendicular to this
line from a given point.
THE PLANIMETER.
143. The Planimeter is an instrument used for measuring
areas that have been drawn to scale. It is a marked example
of high mathematical analysis embodied m a very simple and
useful mechanical appliance. Three oi the best forms of the
mstrument will be described.
F10.3S.
Fig* 35 's Amsler's Polar Planimeter. It consists of a metal
arm, ei, carrying a needle point at ^and pivoted at / to a frame
through which slides a second metal arm, h, and to which is at-
tached an axis, ab, carrying a rolling wheel, c, and a worm gear
which turns a record disk, /. The arm // may be adjusted so
that any required length, within the limits of the instrument,
from pivot i to tracing point d may be used.
When in use the instrument rests on three points, e, d, and
the circumference of the wheel c. To measure an area, the
needle point e is fastened in the drawing in a convenient
position, and the point ^made to circumscribe the required area.
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144
SURVEYING.
This causes the wheel c to rotate, and the number of revolu-
tions made, as indicated by the record disk / and the ver-
nier w, multiplied by a constant, is the required area. The
determination of this constant, involving the theory of the
instrument, will be given in such form as to be intelligible to
students who have not studied the calculus.
Fig. 36.
144. Theory of the Polar Planime-
ter.* — In Fig. 36 the essential parts of
the instrument are lettered as in Fig. 35.
The instrument is so constructed that
the angle q> can never be less than o** nor
more than 180° ; that is, neither d nor c
can cross eu
Any infinitesimal portion of cCs path,
in circumscribing an area, as dd\ may be
conceived to be the resultant of two
infinitesimal component motions, ds and
sd' ; ds being described by a motion of
d about ^ as a center, the angle q) remaining fixed, while sd^
is motion of d directly toward ^, or normal to ds^ the angle q)
changing in value. Each of these motions has its due effect in
moving the wheel c and causing it to turn. For a given motion
of either class the amount the wheel c will turn depends on
the value of the angle ^, as will presently be shown. It is evi-
dent that, in circumscribing a closed area, the tracing point
will move as much toward e as it AoQsfrom e, and that for each
element of motion toward e there will be a corresponding ele-
ment from e with an equal value of <p. Hence the resulting
turning of the wheel for radial motion of d is nil and need
not be considered.
* The following demonstration has been given the author by Prof. Wm. G
Raymond, Rens. Poly. Inst., formerly of the University of Caliifornia.
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ADJUSTMENT, VSE, A^D CARE OP INSTRUMENTS. 14^
Corresponding right and left elements of circumferential
mo'lion of d are not made with equal values of ^, and hence
the final record of the wheel is due to this class of motion.
To show the effect of changes in (p (Fig. 36a) :
When d^ is so situated that ecd
is a right angle, motion about ty ^^-'"'^
<p remaining fixed, causes no roll-
ing of the wheel c. For c rotates
about ^ as a center with radius
e c, which is normal to the axis
of the wheel, and hence the axis
of the wheel lies in the direction
of motion and the wheel slips.
The circumference that would
thus be traced by d, q> remaining
fixed, in a complete revolution about e, is called the zero cir-
cumference. Its radius ^^^ is easily shown to be
i? = V7'+A' + 2^A,
and
(p = cos"' ^.
For (p less than this value, /. e., d farther from ^, right-handed,
or clockwise, motion of d about e will cause c to partly slip and
partly roll, and a moment's consideration will show that, looking
from c to dy its roll will be clockwise. Rolling in this direction
is Cd^^A positive and the wheel is accordingly graduated.
If, on the other hand, d be brought nearer to e than the zero
circumference, clockwise motion of d about e causes c to both
slip and roll, and the roll is counter clockwise, or negative.
Left-handed motion of d will cause ^*s roll to be negative for
d outside t\iQ zero circumference and positive for d inside the
zero circumference.
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146
SURVEYING.
To determine the relation of the roll of the wheel to
the area circumscribed (Fig. 36^) :
Let d d' h^ an infinitesimal cir-
cumferential component of d's mo-
tion due to the motion of the whole
instrument about e through the
infinitesimal angle A.
The wheel c moves through the
arc c c\ partly rolling and partly
slipping. The roll is that component
of its motion cd normal to its axis.
This component may be considered,
in the infinitesimal triangle cc's^ to
be cSy normal to c'i.
Let P be the arc corresponding to
the small angle A when the radius is unity. Then
Fig. 366.
and the roll of the wheel is
cs = P. r*^ . cos (fc$.
Since the angle A is very small, cc' may be considered nor
mal to c'Cy hence
c*cs = e (fVy
e V being a perpendicular on ic' produced.
Then
c'v =^ ec' . cos e(fv
= ^r' cos c*cs.
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ADJOsTMENT, OSE, ANJ> CAJtE OP^ INSTRUMENTS. 147
Whence, cs = P. c'v.
Also, c'v z=fcos q>''g*
Whence, cs = P{fcos q) ^ g). (i)
And this is the roll of the wheel due to motion of d through
dif.
To get an expression for the area dd'o'ox
By Trigonometry,
ed^ V/* + //' + 2 fh cos q>9
dd' = ed. P.
The area of a sector is its arc multiplied by half its radius ;
whence,
Area fdd'=iP(/^+/^ + 2/A cos (p). (2)
Similarly, using the value of eo previously founds
A^TC^LCod = i P{r+ h' + 2gh). (3)
Subtracting (3) from (2), it is found that
Area dddo = Ph {f cos ^— ^), (4)
which, it is observed, is equation (i) multiplied by A, whence
the
Proposition. — The distance rolled by the wheel c, for given
motion of d^ when multiplied by //, gives the area included be-
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14$
SURVEYING.
i7veen ds path, the zero circumference, and the radial lines to d*s
initial and final position.
To show that in circumscribing a closed area with the trac-
ing point d the roll of the wheel is correctly summed for
motion of d right handed and left handed, and inside or out-
side the zero circumference :
In Fig. 36^, let the area dd^d^d^ be traced, beginning at d,
and moving clockwise. For mo-
tion from d to d^ the wheel rolls
positively an amount equal to the
area ddfi'o divided by h ; for mo-
tion from d^ to d^ the roll of the
wheel will be neutValized by mo-
tion from d^ to d\ for motion from
rf, to d^ the roll of the wheel is
negative and equal to the area
d^o'od^ divided by h. Hence the
resulting roll of the wheel due to
circumscribing the area dd^d^^ is
that area divided by A.*
Since the area obtained by the wheel for a given motion of
the tracer d is the area lying between d's path and the zero cir-
cumference, it follows that if e is placed within the area to be
measured so that d makes a complete revolution about e there
must be added to the result obtained,' paying attention to its
sign, the area of the zero circumference.f It is
^ = ;r(/« + A' + 2A^);
♦ Let the student reason similarly of the areas d%d^dAd% and d^idndt,
4 Let the student discuss with a diagram the three cases : i. Perimeter wholly
without the zero circumference ; 2. Perimeter wholly within the zero circumfer-
ence ; and, 3. Perimeter partiy within and partly without the zero circumference.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 149
k being known, ifyand^are not furnished by. the maker, the
area may nevertheless be found by circumscribing a known
area with the point e within it and comparing it with the
result obtained by the instrument.
Since the roll of the wheel for each area circumscribed
must be multiplied by k to obtain the area, and since the roll
is equal to the circumference of the wheel multiplied by the
number of revolutions (the quantity read from the scale), it
becomes necessary to determine values for h for the various
kinds of areas that may be measured, such that the work of
multiplying may be a minimum.
145. To Find the Length of Arm for Given Conditions.—
Let c be the circumference of the wheel and n the number of
revolutions for a given area, A ; then
A ^ knc.
Since ^ is a fixed quantity, k c may be placed equal to a con-
venient constant and a value for h determined. The only
caution to be observed is that h must be made a convenient
length for use. If it is desired to measure square inches,
let he ^ \o\
whence, h = —
Since c is usually between two inches and three inches, this
will give a convenient value for A, which value being set, any
area circumscribed by d is given by
A = ion.
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150 SURVEYING,
The reading of the instrument is usually to one one-thou-
sandth of a revolution, the disk recording whole revolutions
and the scale and vernier fractional revolutions.
The value of c is usually furnished by the maker, but for an
additional charge. If not so furnished, it may be found thus:
Circumscribe any known area, with e outside of it, with a
known length of arm A, and note the record «, of the wheel.
Then ^ = 7 — •
hn
If the arm k is not graduated, it may be set by trial so that
A = \on when the value of c is not required. If a diagram be
drawn to a scale of/* feet per inch, each square inch of paper
represents/' square feet of actual area ; and if it is desired to
avoid multiplication, the arm h may be set so that a single
revolution of the wheel shall correspond to a convenient
number of units of actual area. Let A be the actual area
represented by a on the diagram ; then
Pa = A,
and a = hnc\
whence, A = /'A n c.
phc may be assumed a convenient quantity and h deter-
mined.
Example. — The scale of a diagram is 40 feet per inch ; then
A = 1600 hnc.
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ADJUSTMENT, USE, AND CARE OF INSTRifMENTS, 151
Assume i6cx)Ar = 20000;
20000.
then h =
1600^*
which will give a practical working value for A, and th#
figures representing the record of the wheel for any area cir-»
cumscribed, without decimal point, multiplied by 20, will giv^
the area in square feet. The instrument is supposed to read
to thousandths of a revolution. If the horizontal scale is J
feet per inch and the vertical scale is/' feet per inch, the equa-
tion for determining h would stand
A =/fhnc.
Similarly, if a number of areas are to be determined and multi
plied by a constant length, d, as in railroad earth-work, the
expression for a single volume v, of area a, would be, dividing
by 27. to reduce to cubic yards,
but
whence,
A convenient quantity may be chosen for - — A^and h deter-
mined.
The sum of a series of volumes, each of length //, is
r= t;, + z/, + z/, + &c = 2v
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*52
SURVEYING,
= 2 <- hnc
27
= - — Ac2n;
27
whence it is only necessary to circumscribe each area in sue
•cession, add the records of the wheel, and apply the coefficient.
If all the areas are platted from the same base-line the tracer
may be moved continuously around the areas, traversing each
area as many times as it is used in the summing, and only the
final single record read, which is of course the sum of all the
records. A great variety of other problems may be solved
with the instrument.
146. The Suspended Planimeter.— This is shown in Fig.
37. It is essentially a polar planimeter, the pole being at C.
F^c 37.
It has the advantage of allowing the wheel to move over the
smooth surface of the plate 5, instead of over the paper, thus
giving an error about one sixth as great as that of the ordina-
ry polar instrument. The theory of its action is essentially
the same as the other.
147, The Rolling Planimeter is the most accurate instru-
ment of its kind yet devised. Its compass is also indefinitely
increased, since it may be rolled bodily over the sheet for any
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 153
distance, on a right line, and an area determined within certain
limits on either side. It is therefore especially adapted to the
measuring of cross-sections, profiles, or any long and narrow
surface. Fig. 38 shows one form of this instrument as de-
signed by Herr Corradi of Zurich. It is a suspended planim-
eter, inasmuch as the wheel rolls on a flat disk which is a
part of the instrument, but it could not be called a polar pla-
nimeter, the theory of its action being very different from that
instrument. The frame B is supported by the shaft carrying
C Ci
Fig. 38.
the two rollers i?,. To this frame are fitted the disk A and the
axis of the tracing-arm F, The whole apparatus may thus move
to and fro indefinitely in a straight line on the two rollers while
the tracing-point traverses the perimeter of the area to be
measured. The shaft carries a bevel-gear wheel, R^, which
moves the pinion i?,. This pinion is fixed to the axis of the
disk, and turns with it, so that any motion of the rollers R^
causes the disk to revolve a proportional amount, and the
component of this motion at right angles to the axis of the
wheel E is recorded on that wheel. If the instrument remains
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J 54
BVRVEvmc.
stationary on the paper (the rollers R not turning) and the
tracing-point moved laterally, it will cause no motion of the
wheel, since its axis is parallel to the arm F, and turns about
the same axis with F, but 90° from it ; the wheel £, therefore,
moves parallel with its axis and does not turn.
148. Theory of the Rolling Planimeter.— This will be
developed by a system of rectangular coordinates, the path of
the fulcrum of the tracing-arm being taken as the axis of
Fig. 39.
abscissae. The path of the tracing-point will be considered
as made up of two motions, one parallel to the axis of abscis-
sae and the other at right angles to it. The elementary area
considered will be that included between the axis of abscissae
and two ordinates drawn to the extremities of an elementary
portion of the path. But since this element of the perimeter
is supposed to be made up of two right lines, one perpendicu-
lar to the axis of abscissae and the other parallel to it, our
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 155
elementary area must also be divided in a similar manner.
It will at once be seen that one part of this area is zero, since
the two ordinates bounding it form one and the same line.
This is the part generated by the motion at right angles to
the axis of abscissae. Now, we have just shown in the previous
article that the wheel-record for this part of the path is also
zera* We are brought therefore to this important conclusion :
that all components of motion of the tracing-point at right angles
to the axis of abscissce have no influence upon the result. We
will therefore only discuss a differential motion of the tracing-
point in the direction of the axis of abscissae.
In Fig. 39, which is a linear sketch of the instrument shown
in Fig. 38, with the corresponding parts similarly lettered, it
is to be shown that the motion of the wheel E caused by the
movement of the tracing-point over the path dx is equal to
the corresponding dirtdi ydx multiplied by some constant which
is a function of the dimensions of the instrument.
It is evident that a motion of the tracing-point in the di-
rection of the axis of abscissae can only be obtained by moving
the entire instrument on the rollers by the same amount, and
therefore when the point moves over the path dx the circum-
ferences of the rollers R^ have moved the same amount. This
causes a movement of the pitch circle of R^ of dx -^'. This
motion is conveyed to the disk through j?„ so that any point
on this disk, as a, distant ad from the axis, moves through a
* This is not strictly correct, although it leads to correct results. The compo-
nents of motion are really one parallel to the axis of x, and one about the pivot g
as a center. This latter movement makes no record on the wheel, but it does gen-
erate an elementary area included between this element of the path, the ordinates
to its extremities, and the axis of x. These areas are either added to or cut from
tiie areas made by the onward element of motion in such a way as to exactly balance
Wiien the tradng point starts from and returns to the axis of jt, or when it closes on
the initial point, wherever it may be.
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15^ SURVEYING,
distance equal to dx -^ , — . Let ab^ Fig. 39, be this distance,
Then we have .,.,..
''^='^''%-% ^'>
The motion of that portion of the disk on which the roller
rests, equal to ab, causes the circumference of the wheel E to
revolve by an amount equal to the component of the distance
ab perpendicular to the axis of the wheel. This component
part of the disk's motion is bcy and this is the measure of the
wheel's motion. It therefore remains to show that bc=zydx
multiplied by an instrumental constant.
Now, be = ab sin bac (2)
But bac = a -(- A since gac and bad are both right angles.
Also, bac = supplement of dag = a-\- /3,
Also, from the triangle dagy we have
sin dag : sin agd :: D : ad^
. ,. ^, Z> sin or
or sin{a + /3) = —^j— (3)
Since Fga is also a right angle, we have the angle formed
~F'
y
by Fg and the axis of abscissae equal to or, whence sin a =
We may now write :
6c = ab sin {a + /3) = ab — ^5^ = ^*^pT^- • (4)
Now, substituting the value of ab from (i), we have
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 157
Since D, /?„ F, R^, and R^ are all constants for any one
instrument, we see that the wheel-record is a function of the
area generated by the tracing-point and the instrumental con-
stants, which was to be shown. It now follows that the sum-
mation of all these elementary areas included between the
path of the tracing-point and the axis of abscissae^ is repre-
sented b>' the total wheel-mavement, or the difference between
its initial and final readings, provided the tracing-point starts
from and closes on the axis of ;r, or closes on the starting-point,
wherever that may be.
The following comparison between the rolling and the polar
planimeters may be made : The axis of x corresponds to the
zero circle ; the unrecorded motion about the pivot g. Fig. 39,
corresponds to the balanced record of the motion about r, Fig;
36 ; and the significant forward motion of the former to the
motion about P as a center, in the latter.
As in the case of the polar instrument, the proper length of
arm F, to be used with the rolling-planimeter to give results
in any desired unit, depends on the other instrumental con-
stants. These being known, the value of Ftnay be computed
in the same manner as with the polar planimeter.
149. To test the Accuracy of the Planimeter, there is
usually provided a brass scale perforated with small holes. A
needle-point is inserted in one of these and made fast to the
paper or board, while the tracing-point rests in another. The
latter may now be moved over a fixed path with accuracy.
Make a certain number of even revolutions forward, or in the
direction of the hands of a watch, noting the initial and final
readings. Reverse the motion the same number of revolutions;
and see if it comes back to the first reading. If not, the dis-
crepancy is the combined instrumental error from two meas-
urements due to slip, lost motion, unevenness of paper, etc.
If this test be repeated with the areas on opposite sides of
II
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158 SURVEYING.
the zero-circle in the case of the polar-plan imeter, or on oppo-
site sides of the axis of abscissae in case of the roUing-planime-
ter, with the same score in both cases, it proves that the pivot-
points a, b, ky and the tracing-point ^/(Fig. 35), are in the same
straight line, in case of the polar instrument, and that the cor-
responding points in the suspended and rolling planimeters
form parallel lines; in other words, that the axis of the meas-
uring-wheel is parallel to the tracing-arm. If the results differ
when the areas lie on opposite sides of the axis or zero-circle,
these lines are not parallel and must be adjusted to a parallel
position.
150. Use of the Planimeter. — The paper upon which the
diagram is drawn should be stretched smooth on a level sur-
face. It should be large enough to allow the rolling-wheel to
remain on the sheet.
The instrument should be so adjusted and oiled that the
parts move with the utmost freedom but without any lost mo-
tion. This requires that all the pivot-joints shall be adjustable
to take up the wear. The rim of the measuring-wheel must be
kept bright and free from rust. The instrument must be han-
dled with the greatest care. Having set the length of the
tracing-arm for the given scale and unit, it is well to test it
upon an area of known dimensions before using. If it be found
to give a result in error by — of the total area, the length of the
tracing-arm must be changed by an amount equal to this same
ratio of its former length. If the record made on the wheel
was too small then the length of the tracing-arm must be di*
minished, and vice versa. If the paper has shrunk or stretched,
find the proportional change, and change the length of the
tracing«arm from its true length as just found, by this same
ratio, making the arm longer for stretch and shorter for shrink-
age. Or the true length of arm may be used, and the results
corrected for change in paper.
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ADrVSTMENT, USE, AND CAkE OP INSTRUMENTS, IjQ
To measure an area, first determine whether the fixed point,
or pole, shall be inside or outside the figure. It is preferable
to have it outside when practicable, since then the area is ob-
tained without correction. If, however, the diagram is too
large for this (in case of the polar planimeter) the pole may be
set inside. In either case inspection, and perhaps trial, is nec-
essary to fix upon the most favorable position of the pole, so
that the tracing-point may most readily reach all parts of the
perimeter. If the area is too large for a single measurement,
divide it by right lines and measure the parts separately.
Having fixed the pole, set the tracing-point on a well-defined
portion of the perimeter, and read and record the score on
the rolling-wheel and disk. This is generally read to four
places. Move the tracing-point carefully and slowly over the
outline of the figure, in the direction of the hands of a watch,
around to the initial point. Read the score again. If the
pole is outside the figure, this result is always positive when
the motion has been in the direction here indicated. If the
pole is inside the figure, the result will be negative when the
area is less than that of the zero-circle, positive if greater.
With the pole inside the figure, however, the area of the zero-
circle must always be added to the result as given by the score,
and when this is done the sum is always positive, the motion
being in the direction indicated. The area of this zero-circle
is found in art. 144, to be n {m^ + /' + 2nl). The value
of /, which is the length of the tracing-arm, is known. The
values of m and n should be furnished by the maker. If these
are unknown, the area of the zero-circle can be found for any
length of arm /, by measuring a given area with pole outside
and inside, the difference in the two scores being the area of
this circle. By doing this with two very different values of /
we may obtain two equations with two unknown quantities,
m and «, from which the absolute values of these quantities
may be found. Thus we would have :
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l6o SURVEYING.
A'=^7r{m' + r + 2nl');
whence m^ + 2»/ = /' ;
TV
' TV
wherein /, /', A, and A^ are known. The values of m and n are
then readily found.
In using the rolling-planimeter, it is advisable to take the
initial point in the perimeter on the axis of abscissae, as in this
position any small motion of the tracing-point has no effect
on the wheel, and so there is no error due to the initial and
final positions not being exactly identical.
The planimeter may be used to great advantage in the
solution of many problems not pertaining to surveying. In
all cases where the result can be represented as a function
of the product of two variables and one or more constants, the
corresponding values of the variables may be plotted on cross-
section paper by rectangular coordinates, thus forming with
the axis and endordinates an area which can be evaluated for
any scale and for any value of the constant-functions by setting
off the proper length of tracing-arm. Thus, from a steam-
indicator card the horse-power of the engine may be read off,
and from a properly constructed profile the amount of earth-
work in cubic yards in a railway cut or fill. Some of these
special applications are further explained in Part II. of this
work.
151. Accuracy of Planimeter-measurements.— Professor
Lorber, of Loeben, Austria, has thoroughly investigated the
relative accuracy of different kinds of planimeters, and the re-
sults of his investigations are given in the following table. It
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. l6l
will be seen that the relative error is less as the area measured
is larger. The absolute error is nearly constant for all areas, in
the polar planimeter. The remarkable accuracy of the rolling-
planimeter is such as to cause it to be ranked as an instrument
of precision.
TABLE OF RELATIVE ERRORS IN PLANIMETERMEASUREMENT8.
Aeba
IN—
The error in one passage of the tracer amounts on an
average to the following fraction of the area meas-
ured by—
The ordinary po-
lar plan i meter-
Unit of vernier:
10 aq. mm.
=:.oi5&q. in.
Suspended plani-
meter-Unit of
vernier:
I sq. mm. =
.OCX sq. iu.
Rolliiig planime-
ter-Unit of ver-
nier:
z sq. mm. =
.001 sq. in.
Square cm.
Square inches.
lO
1.55
^
<k
xAlF
20
3.10
xir
tAt
tAit
50
7.75
Tk
kVo
Ww
lOO
15.50
T^
tAt
Wtnr
200
31.00
Thx •
tAt
tAt
300
46.50
....
nW
loiod
THE PANTOGRAPH.
^ 152. The Pantograph is a kind of parallel link-motion
apparatus whereby, with one point fixed, two other points are
made to move in a plane on parallel lines in any direction.
The device is used for copying drawings, or other diagrams to
the same, a larger, or a smaller scale. The theory of the instru-
ment rests on the following :
Proposition : 1/ the sides of a parallelogranty jointed at
the corners A, B, C, and D, and indefinitely extended, be cut by
a right line in four points, as E, F, G, and H, then these latter
points will lie in a straight line for all values of each of the
parallelogram angles from zero to 180°, and the ratio of the dis-
tances EFy FG, and GH will remain unchanged.
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1 62
SURVEYING.
In Fig. 40, let A, B, C, Z?be the parallelogram, whose sides
(extended) are cut by a right line in Fy G, E, and H. It is
evident that one point in the figure may remain fixed while
t/
Fig. 40.
the angles of the parallelogram change. Let this point be G.
Since GC and GHy radiating from 6?, cut the parallel lines
DE and CH, we have
GD\DE :: GC\ CH.
Also, for similar reasons,
ED\DG\\ EA :AF.
Now since the sides of the parallelogram, as well as all the
intercepts, AF^ GD, DE, and CH, remain constant as the angles
of the figure change, when the figure has taken the position
shown by the dotted lines, we still have
also,
GU : D'E wGC \ CH'\
E'U : D'G :: EA' : A'P.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 163
From the first of these proportions we know that G, E\
and H' are in the same straight line, and the same for G, E\
and F ; therefore, they are all four in the same straight line.
To show that they are the same relative distance apart as
before we have,
FG:GE: EH :: BC\DE\ CH-^DE^
also,
FG : GE : EH' :: EC : UE : CE^UE.
But
BC = EC, DE = D'E, and CH^DE^ CH' - EE^
therefore we may write,
FG : GE : EH :: FG \ GE : EE.
Q. E. D.
It is evident that two of the points E, F, G, and H may
become one by the transversal passing through the point of
intersection of two of the sides of the parallelogram. The
above proposition would then hold for the three remaining
points.
In the Pantograph only three of the four points E, F,
G^ and H (Fig. 40) are used. One of these may therefore
be taken at the intersection of two sides of the parallelogram,
but it IS not necessarily so taken. These three points are: the
fixed point, the tracing-point, and the copying-point.
In Fig. 41, F\s the fixed point, held by the weight P\ B is
the tracing-point, and D is the copying-point, or vice versa as
to B and D. The parallelogram is E, G, B, H. The points
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164
SURVEYING,
F, By and D must lie in a straight line, B being at the inter-
section of two of the sides of the parallelogram. The points
Ay Ey and C are supported on rollers. In Fig. 42. the fixed-
Fig. 4x.
point is the point of intersection of two of the sides of the
parallelogram. The upper left-hand member of the frame is
not essential to its construction, serving simply to stiffen the
Fig. 4*.
copying-arm, the fourth side of the parallelogram being the
side holding the tracing-point.
In Fig. 43, neither of the three points is at the intersection
of two of the sides of the parallelogram, and hence there is a
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 165
fourth point unused, having the same properties as the fixed,
tracing, and copying points, it being at the intersection of the
line joining these three points with the fourth side of the par
allelogram.
From the theoretical discussion, and from the figures shown,
it becomes evident that there may be an indefinite variety of
Fig. 43.
combinations which will do the work of a pantograph. The
only essential conditions are that the fixed, the tracing, and the
copying points shall lie in a straight line on at least three sides
of a jointed parallelogram, either point serving any one of the
three purposes.
153. Use of the Pantograph. — The use of the instrument
is easily acquired. Since both the tracing and copying points
should touch the paper at all times, such a combination as that
shown in Fig. 41 is preferable to those shown in Figs. 42 and
43, since in these latter the tracing point is surrounded by sup-
ported points, and so would not touch the paper at all times
unless the paper rested on a true plane. In most instruments
where the scale is adjustable, the two corresponding changes
in position of tracing and copying points for different scales
are indicated. To test these marks, see that the adjustable
points are in a straight line with the fixed point, and to test the
FD
scale see that the ratio ^- (Fig. 41) is that of the reduction
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1 66 SU/^VEYING.
desired. Thus, if the diagram is to be enlarged to twice the
original size, make JFD = 2FB ;
DE FE
or make -ftt; ="n^ir = scale of enlargement.
If the drawing is to be reduced in size, make j9 the copying-
point and D the tracing-point.
If the drawing is to be copied to the same scale, make BF
= ED and make B the fixed point. The figure is then copied
to same scale, but in an inverted position.
In the best instruments the arms are made of brass, but
very good work may be done with wooden arms.
PROTRACTORS.
154. A Protractor is a graduated circle or arc, with its cen-
tre fixed, to be used in plotting angles. They are of various
designs and materials.
Semicircular Protractors, such as shown in Fig. 44, are
usually made of horn, brass, or
german-silver. They are grad-
uated to degrees or half-degrees,
and the angle is laid off by
holding the centre at the vertex
of the angle, with the plain
edge, or the o and 180 degree
' ' ^ line on the given line from which
the angle is to be laid off.*
In the full circle protractor, shown in Fig. 45, there is a
movable arm with a vernier reading to from I to 3 minutes.
The horn centre is set over the given point, the protractor
oriented with the zero of the circle on the given line, and the
arm set to the given reading when the other line may be
drawn.
* For an elegant style of protractor to be used in toijographical work, see Fig
66, p. 273.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. 167
The threerarnt protractor. Fig. 46, has one fixed and two
movable arms by which two angles may be set off simulta-
neously. It is used in plotting observations by sextant of two
Fig. 4S.
angles to three known points for the location of the point of
observation. This is known as the three-point problem and
is discussed in Chap. X.
Fig. 46.
Paper protractors are usually full circled, from 8 to 14
inches in diameter, graduated to half or quarter degrees.
They are printed from engraved plates 011 drawing- or tracing-
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i68
SURVEYING,
paper or bristol-board, and are very convenient for plotting
topographical surveys. The map is drawn directly on the
protractor sheet, the bearing of any line being taken at once
from the graduated circle printed on the paper. These " pro-
tractor sheets" can now be obtained of all large dealers.
The coordinate protractor* xs^. quadrant, or square, with
Fig. 47.
angular graduations on its circumference, or sides, and divided
over its face by horizontal and vertical lines, like cross-section
paper. A movable arm can be set by means of a vernier to
read minutes of arc, this arm being also graduated to read
distances from the centre outward. Having set this arm to
read the proper angle, the latitude is at once read off on the
* Called a Trigonometer by Keuffel & Esser, the makers.
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS. I69
vertical scale and the departure on the horizontal scale for the
given distance as taken on the graduated arm. A quadrant
protractor giving latitudes and departures for all distances
under 2500 feet to the nearest foot, or under 250 feet to the
nearest tenth of a foot, has been used. The radius of the cir-
cle is i8i inches. Both the protractor and the arm are on
heavy bristol-board, so that any change due to moisture will
affect both alike and so eliminate errors due to this cause.
The instrument was designed to facilitate the plotting of the
U. S. survey of the Missouri River.* It has proved very
efficient and satisfactory. A similar one on metal, shown in
Fig. 47, is now manufactured, and serves the same purpose.
PARALLEL RULERS.
155. The Parallel Ruler of greatest efficiency in plotting
is that on rollers, as shown in Fig. 48. The rollers are made
of exactly the same circumfer- «
ence, both being rigidly attached
to the same axis. It should be L
made of metal so as to add to its ^'°- 48.
weight and prevent slipping. It is of especial value in connec,
tion with the paper protractors, for the parallel ruler is set on
any given bearing and then this transferred to any part of the
sheet by simply running the ruler to place. Two triangles
may be made to serve the same purpose, but they are not so
rapid or convenient, and are more liable to slip. The parallel
ruler is also very valuable in the solution of problems in
graphical statics.
SCALES.
156. Scales are used for obtaining the distance on the
drawing or plot which corresponds to given distances on the
* For sale by A. S. Aloe & Co., St. Louis, Mo,
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170
SURVEYING,
object or in the field. There is such a variety of units for
both field and office work, and a corresponding variety of
scales, that the choice of the particular kind of scale for any
given kind of work needs to be carefully made. Architects
usually make the scale of their drawings so many feet to the
inch, giving rise to a duodecimal scale, or some multiple of ^.
^ I ^^ 1 ll' I V 1 !^ 1 ^ 1
liiiliiliiilllllllllTlllllIM ''MlllllMilllllIM Mllllll
h— ^
0
9
?
7
«
J
k
». .
»
»
a 4 « %
* 8
2
tl^
S
—
=r:
—
—
^
M
a
«
a
2:
LJ
LJ
LJ
LJ
LJ
LJ
LJ
^ ••
^1^
^
Fig. 49.
A surveyor who uses a Gunter's chain 66 feet in length plots
his work to so many chains to the inch, making a scale of
some multiple of tj-^Vit- ^^ engineer usually uses a 100-foot
chain and a level rod divided to decimal parts of a foot ; so he
finds it convenient to use a decimal scale for his maps and
drawings, reduced to the inch-unit however. Here the field-
unit is feet and the office-univ is inches, both divided deci-
mally. This gives rise to a sort of decimal-duodecimal system,
the scale being some multiple of ^Jt. Various combinations
of all these systems are found.
Figure 49 shows one form of an ivor}' scale of equal parts
for the general draughtsman. The lower half of the scale is
designed to give distances on the drawing for 4, 40, or 400
units to the inch when the left oblique lines and bottom
figures are used, and for 2, 20, or 200 units to the inch when
the right oblique lines and top figures are used. Thus, if we
are plotting to a scale of 400 feet to the inch, and the dis-
tance is 564 feet, set one point of the dividers on the vertical
line marked 5, and on the fourth horizontal line from the bot-
tom. Set the other leg at the intersection of the sixth inclined
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ADjaSTMEh/T, VSt, At^b CAkE OP INSTRUMENTS, l^t
line with this same horizontal line, and the space subtended by
the points of the dividers is 564 feet to a scale of xiv^-
Figure 50 is a cut of an engineer's triangular boxwood
scale, 12 inches long, being divided into decimal inches.
There are six scales on this rule, a tenth of an inch being sub-
divided into I, 2, 3, 4, 5, and 6 parts, making the smallest
Fio. so.
graduations -^y ^, ^, -j^y -^, ^^ ^^ ^" >"ch respectively. This
is called an engineer's or decimal-inch scale The architect's
triangular scale is divided to give \, J, f , J, }, I, \\, 2, 3, and
4 inches to the foot. Such a scale is of less service to the
civil engineer.
THE POCKET SLIDE RULE.*
X56a. The Slide Rule in some of its various forms is by far the most easy
and rapid means of carrying on computations by multiplication and division that
has ever been devised. One can work with it continuously without becoming men-
tally weary. The author, having had a large experience in the use of all kinds of
mechanical computing machines and of logarithmic and multiplication tables, is
fully convinced that the slide rule far exceeds them all in the ease, rapidity, and
accuracy with which it can be used by any one after an hour's instruction or prac-
tice. While this little implement is in almost universal use in Europe, not only
by scientific but by all classes of practical men, it has never come into general use
in America. This is greatly to be deplored, and it is hoped our deficiency in this
respect will now be remedied, by the placing of a cheap but accurate pocket slide
rule upon the American market.
The rule in this pocket form (Fig. sew) gives results to three significant figures,}
♦ While the use of the slide rule has no special relation to surveying, its use
may properly be described here as one of the instruments of engineering com-
putation.
• f This rule can now be purchased in America for $1.00, but a better one,
printed on celluloid, and made adjustable so as to obviate all objections on the
score of swelling, warping, or looseness from wear, is now manufactured by the
Keuflfel Esser Co., New York.
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SUR^EYIt^G.
which is surficiently accurate for most of the computations which have to be made
in all kinds of physical investigations, and in the solution of engineering problems.
Where four or live signiHcant figures are required, Thacher's slide rule (Fig. 50^^) will
be used, but for a large part of the work for which this larger slide rule is employed
Fig. 5Qfl.
a small pocket form is really sufficient, since this gives the results to a greater
degree of accuracy than it is possible to obtain in the original data. After a little
use of this instrument, it becomes almost a necessity. It is not too much to say
that it is indeed a great benefactor in preventing the mental fatigue which always
accompanies continued numerical operations in multiplication and division.
The following simple rules and explanations are readily understood and followed
with the slide rule in hand, and they fullv exemplify most of its applications. A
Fig. 50*.
working knowledge of the use of logarithms is assumed. A small magnifying glass
may sometimes be employed to increase the accuracy of the settings and readings.
THE SCALES A AND B.
The scales A and B represent the logarithms of the numbers marked upon
them, from i to 10, or from 10 to 100, or from 100 to 1,000, according to the degree
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, lyib
of accuracy required, to a uniform scale. By the movement of scale B, along scale
A, to the right or left, these logarithms can be mechanically added or subtracted,
or both simultaneously, and multiplication and division thus effected. Computa-
tions made on the slide rule, therefore, are mechanically performed by means of
logarithms represented graphically. The two scales, A and B, are exactly alike,
each containing two duplicate, complete, graphical logarithmic scales.
The slide rule is used to solve a problem in proportion^ or in general to multi-
ply two numbers together and to divide by a third, any one of which may, of
course, be unity, the general problem being to perform the work indicated in the
following : ~r- = ?
Since the scales increase to the right, a movement of the sliding scale B to the
left subtracts a logarithm, and a movement to the right adds one. Thus, to divide
by b we move the slide to the left until b comes under i of the A scale. Then to
multiply by a we move it fon\'ard until b comes under a of that scale. But this is
equivalent to setting b on the B scale opposite a on the A scale, at once. Then
every number on the sliding {B) scale has been multiplied by a and divided by b
relatively to the corresponding number on the A scale. Hence any number on the
sliding scale, as jt, has opposite to it a number which is -r times x. Therefore we
have the rule :
I. To solve -7- set ^ on ^ opposite a on ^, then find x on B, and the opposite
reading on A is the result.
^ .. , 3-04 X 20.1
Example : Solve ~~ — = 1.025.
59.6
Paying no attention to decimal points, treat the quantities as whole numbers,
and setting 596 on B opposite 304 on A, find opposite 201 on B 1.025 on ^- The
decimal point is usually best located by a mental computation or inspection. Evi-
dently the result is somewhat more than unity, hence the point comes after
the I.
When a series of results is to be obtained in which two of the numbers, as a and
b, are constants, and the other varies, always set for the constant terms, after
which the entire series of results can be read off for the varying values of x without
further settings.
THE SCALES C AND D.
The log. scales Cand D are so arranged that the numbers on them are the square
roots of those in the same vertical line in scales A and B. Transfers are taken
from scales C ox D \o scales A ox Bhy means of the visible line on the transparent
sliding indicator.
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17*^ SURVEYING.
Thus if the problem be tj, move the transparent indicator to a on A ^ then move
the sliding scale till ^ on C (which is the same as ^' on B) comes to the line on the
indicator, then look up jr on i? and find the result on A as before. This is the same
as the former case, except b is now taken on the C, the square root scale, in place
of b^ on the B scale.
Example : ** , ^ \ '^ = 0.308.
(24.2)'
Setting the line of the sliding indicator on 469 of the A scale, bring 242 on
sliding scale C under same line. Then look up 385 on scale B and find 308 on A.
The decimal point is located by a mental computation as before.
If the problem be — , move the indicator to a on Z>, then bring b on the sliding
scale B to the line, and look up j; on ^ as before, finding the result again on A.
„ , (98.i)» X 63.8
Example : — — =^ = 1360.
452
Here we set the sliding indicator on 981 on /? (which is the position for (981)*
on ^), and bring 452 on sliding scale B to line, then looking up 638 on B find 136
on A. The location of the decimal point is made by a hasty approximate mental
computation. Thus, (98.1)" is nearly 10,000, and 63.8 will go into 452 about 7
10,000 . ^ TX , . . ^
times. IS about 1400. Hence the result is 1360.
To find the square or square root of a number simply set the sliding indicator
on the number, finding this on A if the square root is sought, and on D if the square
is to be found, the result being then under the same line on the other scale. In
finding square roots use only the left-hand half of scale A for an odd number of
figures to the left of the decimal point, and the right-hand half for an even numbei
of figures before the decimal point.
Examples : Find the square of 34.8.
Setting the line of the transparent indicator on 348 of scale /?, it is found to
coincide with 121 of scale A, The answer is evidently 1 2 10, to the nearest reading
of the rule.
Find the square root of 7. 56.
Set the line of the indicator on 756 of first half of scale A^ and find under sam«
line 275, the answer bein^ evidently 2.7$.
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ADJUSTMENT, USE. AND CARE OF INSTRUMENTS. 17 \d
Find the square root of 13.9.
Set the line of the indicator on 139 on right half of scale A^ and find below D
3.72 as the answer.
To find tJu cube of a number we have /»* = . Hence set i on B^ opposite
a on />, then for a on ^ we find a* on ^.
Example ; Find the cube of 6.48.
Bring line of transparent indicator over 6.48 on Z>, and bring i on ^ to this
mark. Then opposite 6.48 on ^ find 272 on ^.
To find the cube root we reverse the process.
t 3^ X X
Thus i^a = JT, wherein = a. Here we must find an unknown number
such that when it falls on B under the given number on A^ it is also found on D
opposite unity (right or left) on C.
Example : Find the cube root of 456.
Knowing approximately, by inspection, that it is in the vicinity of 8, we bring
8 on ^ under 456 on A (putting first the line on the transparent indicator at 456 on
A for convenience) ; we then find that the right-hand index, or unity, on C falls a
little to the left of 8 on D. By a little adjustment of the sliding scale we find that
when 770 on B comes to the line (456 on A) that 770 is also opposite unity on C at
the right end. The answer is therefore 7.7.
If the number had been 45.6, we should have had to use the right-hand scale of
A and the left-hand index of C, thus finding the answer to be 3.57.
ofHx ap'
For carrying on continuous computations^ as '?^. we may take -^ first, and
by setting ^ on i9 to a on ^, move the transparent indicator to g on B. Then set
y on ^ to this line and move indicator to h on B. Then set ^ on ^ to this line
again, and look up x on B^ opposite to which is the answer on A,
Thus. '■42x16.9x64x0132 ^
42.9 X 0.046 X 3.26
Here we set 429 on B opposite 242 on A, and move line on indicator to 169 on
B. Then move 46 on B to line, and set indicator on 64 on B. Then move 326 on
B to the line, and look up 132 on B. Opposite this find the answer 537 on ^. To
find the position of the decimal point we perform a mental computation like this :
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171^ SURVEYING.
2.42 X 16.9 = about 40 ; 0.132 = about f, and this into 64 is about 9 ; 9 x 40 =
360 ; 0.046 is about fV, and this into 42.9 is about 2 ; 2 x 3.26 is about 7 ; f of 360
is about 50 ; hence the answer is 53.7, rather than 5.37 or than 537.
It goes without saying that any one or more of these numbers could have been
a square, in which case scale C is used in place of B and scale D in place of A.
TRIGONOMETRICAL COMPUTATIONS.
On the under side of the slide three scales will be found, the upper one marked
Sy being a scale of natural sines, and the lower one marked T^ a scale of natural
tangents. Between these is a scale of equal parts, which gives the logarithms cor-
responding to the series of numbers on scale D,
In order to use these, place the slide in the groove with the under side upper-
most, and the left and right indices coinciding. On A will then be found the sines
of the angles given on 5, those on the left of scale A having the characteristic — i,
and those on the right of scale A^ the characteristic 0, thus we find,
Sine 3* = 0.0523, on left of scale A.
Sine 15* 10' = 0.262, on right of scale A,
We have on D the tangents of the angles given on T', the characteristic being
always o ; thus, tan. 25° = 0.466 on scale D. The scale g^ves the tangents from 6
to 45 degrees only ; for angles less than 6" the tangent is practically the same as
the sine ; larger angles must be found by the formula,
tan a =
tan (90 — tf)'
The sines and tangents of angles may be found without reversing the slide, by
setting the given angle on scale 5 or T' to the index line on the transparent disk on
the under side of the rule, and reading off the sine on B or tangent on C opposite
the right-hand index marks of the scales A and D.
T/ie Logarithms of numbers are found in a similar manner by setting the left
index of C to the given number on /?, and reading off the logarithms on the scale
of equal parts under the index on the reverse side of the rule.
With the scale of equal parts the cube and other roots or powers may be extracted,
b 1.7
such as jr*, JT , etc.
Example : Find 4* or 4/4*.
By the above method log 4 = .602, and log 4* = .602 x J = 1.505. Now by
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ADJUSTMENT, USE, AND CARE OF INSTRUMENTS, 17 \f
placing 505 on the log. scale on the lower side of the slide at the index line on the
under side of the rule, we find 32 on scale D under the left index of C, which is
therefore equal to 4^, the logarithmic index being i.
The position of the decimal point in all these cases will be easily ascertained by
those accustomed to this class of calculations.
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BOOK il.
SURVEYING METHODS.
CHAPTER VII.
LAND-SURVEYING.^
157. Purposes. — All surveys of land, properly so called
are made
{a) For the purpose of establishing certain monuments,
corners, lines, and boundaries, as in laying out and dividing
land, or,
{b) For the purpose of identifying and locating such monu-
ments, lines, and boundaries, after they have been established,
as in all resurveys for location and area.
In all cases the boundary and dividing lines are the traces
of vertical planes on the surface of the ground, and the area is
the area of the horizontal plane included between the bound-
ing vertical planes. In other words, the area sought is the
area of the horizontal projection of the real surface.
158. In laying out Land the work consists in running
the bounding and dividing lines over all the irregularities of
the surface, leaving such temporary and permanent marks as
the work may demand. These lines to lie in vertical planes,
and their bearings and horizontal distances to be found. The
bearing of a line is the horizontal angle it makes with a merid-
ian plane through one extremity, and its length is the length
of its horizontal projection. This reduces the plot of the work
to what it would be if the ground were perfectly level. If all
* See Appendix G for the essential requirements of a survey and ina»*
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LAND SURVEYING, 1/3
the straight lines of a land survey lie in vertical planes, and if
their bearings and horizontal lengths are accurately deter-
mined, then as a land survey it is theoretically perfect, what-
ever the purpose of the survey may be.
The needle compass and Gunter's chain have been univer-
sally and almost exclusively used in land surveying. Except
in those localities where there is local attraction and very
erratic changes in the declination of the needle, the compass
is the best instrument that can be used for the purpose. Most
of the trouble which has resulted from its use has arisen from
a failure to make frequent determinations of the declination.
An observation on Polaris by the method given in Art. 33, or
by the use of Table XII., and its description in Art. 381^,
should be made in each township, and a true meridian marked
on the ground at every county-seat. The compass should be
set up on this meridian as often as once a year, at about 10
o'clock A.M., and the declination noted. The annual change
in declination found at the county-seat could then be attrib-
uted to each of the declinations found in the several town-
ships of the county, and so a continuous corrected record of
the true declination kept for all parts of the county.
159. Monuments. — All marks of whatever description left
on or near the surface of the ground, such as stones, stakes,
mounds, holes, or trees specially marked and described, for the
purpose of designating a particular point on the surface as a
landmark, are called monuments.
All land monuments set by surveyors should be stones, suit-
ably cut and marked, and planted in the ground. Surveyors
cannot insist too strongly on the necessity of setting permanent
monuments to mark land boundaries at the time these bound-
aries are first established. The surveyor who first lays out the
ground should always set permanent monuments before the
survey passes beyond his own control. It will not do to trust
that some one interested will replace his temporary marks by
those of a more permanent character afterwards, both because
12
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174 SURVEYING,
this is likely to be entirely neglected, and also because of the
cloud thus thrown over the authority of the monument itself
from the fact that it is not the original mark.
Monuments are used not only to mark corners of tracts of
land, but also to mark points in straight lines, as in State
boundaries, and points fixed by triangulation in geodetic, geo-
logical, State, and municipal surveys.
i6o. Significance and Authority of Monuments. — When-
ever monuments are placed in any scheme of land subdivision,
and these monuments are described in the conveyance of such
lands when sold, they thereby acquire a perpetual and controll-
ing significance. It matters not how frail and temporary a
monument may have been — a mere peg stuck in the ground —
if it did at the time designate a particular point in the boundary
of the tract, and if such monument is recognized in the deed, its
position controls the location absolutely. In any subsequent
survey for the location of the boundary it becomes supremely
important to identify with certainty the true position of such
monument. The field notes of the original survey, or any de-
scription of the boundaries in the deed, or the area called for,
have no weight in determining the position of the lines and
corners as against the certain identification of the monuments
also recognized in the conveyance. What the conveyer sold
and the purchaser bought was a certain fixed tract of land
which should have been marked at one time by visible monu-
ments. In this case the field notes are material evidence of
the original position of the monuments, but since errors in
surveying are not uncommon, and since the supposed area of
the tract is computed from these field notes, neither the area
nor the description by course and distance, called for in the
deed, are allowed to hold as against the proved location of the
original monuments, also called for in the deed.
Surveys are always subject to revision and correction. A
monument once set and used in a conveyance cannot be changed,
even though its position is not what it was intended to be, or
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LAND SURVEYING. 1/5
not what it is said to be,* in the written description, without the
free consent of all parties concerned. There is therefore an
inviolableness and absoluteness of control in recognized monu«
ments which does not pertain to any surveys or*to any de-
scriptions or areas dependent on surveys.t
i6i. Lost Monuments.! — When monuments have once
been established and used in conveyances and afterward dis-
appear or are lost, they cannot be re-established as an abso-
lute authoritative control by any survey or agreement of sur-
veys. Nothing but consent or acquiescence of all the parties
in interest, or a judgment of the court can replace a lost
monument. Surveys and the judgment of surveyors are valu-
able evidence in determining where the original monument was
placed, but the surveyor has no authority or right to replace
or re-establish a lost monument, or to certify to its position, un-
less he can find such trace of the original monument itself, or of
a witness point, as may serve to identify its position with cer-
tainty. He may then replace it by a more permanent mark,
and by recording a full description of his work the new monu-
ment may be recognized as having all the authority of the
original. But any location of a monument based on the field
'notes of the original survey, even in conjunction with other
well-authenticated monuments a considerable distance off,
cannot serve to ** establish ** such monument. It serves only as
so much evidence, to be taken in connection with all other evi-
dence, material and personal, such as fence lines, acknowledged
boundaries, testimony of witnesses, etc., which evidence may,
and often does, outweigh the evidence furnished by the sur-
vey. In such a case the surveyor is an expert witness, en-
gaged to interpret the original field notes and to find where
they would place the lost monument ; but inasmuch as the
original field notes may not have agreed with the actual posi-
tion of the monument, any number of resurveys or agreement
♦ Sec particular case 3, p. 231.
t See Arts. 302, 303, and 304 in Chap. XII., on City Surveying. Also Ap-
pendix I, p. 734, for the rules of the U. S. General Land Office regarding the
restoration of lost or obliterated corners. Digitized byCjOOQlC
1/6 SURVEYING.
of resurveys cannot of themselves be so conclusive evidence of
its original position as to prevent an appeal to the courts.
The making of resurveys, which is the principal business ol
the land Stirveyor, whether in city or country, consists, there-
fore, largely in the search for and satisfactory identification of
corners, marks, boundaries, and other visible objects which
have all the force and authority of monuments. The proved
experience and degree of expertness and reliability of the par-
ticular surveyor doing the work will, of course, affect the value
of the resurvey as compared with other evidence furnished as
to the monuments themselves.
THE UNITED STATES SYSTEM OF LAYING OUT THE PUBLIC
LANDS.
162. The Public Lands of the United States have in-
cluded all of that portion of our territory north of the Ohio
River and west of the Mississippi River not owned by indi-
viduals previous to the dates of cession to the United States
Government ; also similar portions of the States of Florida,
Alabama, Mississippi, and Tennessee. All of this territory,
except the private claims, has been subdivided, or laid out, in
rectangular tracts bounded by north and south and east and
west lines, each tract having a particular designation, such that
it is impossible for the patents or titles, as obtained from the
Government, to conflict. This has saved millions of dollars to
the land-owners in these regions by preventing the litigations
that are common in the old colonial States, and is one of the
greatest boons of our national Government The system was
probably devised by Gen. Rufus Putnam,* an American officer
in the Revolutionary War. It was first used in laying out the
eastern portion of the State of Ohio, in 1786-7, then called the
Northwest Territory. This was the first land owned and sold
* See a paper read before the Engineers' Club, of St. Louis, by Col. H. C.
Moore, and published in ih^ Journal of the Association of Engineering Societies
Vol. II., p. 282.
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LAND SURVEYING. 1/7
by the national Government. The details of the system have
been modified from time to time, but it remains substantially
unchanged. The following is a synopsis of the method now
used, which is given in detail in the Manual of Surveying In-
structionSy issued by the Commissioner of the General Land
Office, at Washington, D. C, and obtained on application : *
163. The Reference Lines are first a Principal Meridian
and an accompanying Base Line. There have been twenty-
four sets of these meridian and base lines used in laying out
the public lands in different parts of the United States, for a
detailed description of which see Appendix E.
From the principal meridian and its accompanying base
line, guide meridians and standard parallels are run north
and south from the base line and east and west from the prin-
cipal meridian, twenty-four miles apart in each direction.
These lines are run with great care, using the solar compass or
solar attachment. The magnetic needle cannot be relied on
for this work, for two reasons : there may be local attraction
from magnetic deposits, and the declination changes rapidly
(about a minute to the mile) on east and west lines. The
transit alone might be used to run out the meridians, as this
consists simply of extending a line in a given direction. If the
transit is used in running the parallels offsets m.ust be taken
as described in Art. 169, p. 185. The solar compass is the
only surveying instrument that can be used for running a
true east and west line an indefinite distance. The needle-com-
pass would serve if there were no local attraction and if the true
declination were known and allowed for at all points. The solar
compass (or solar attachment) is the instrument for this work.
In running these reference-lines, every eighty chains (every
mile) is marked by a stone, tree, mound, or other device, and is
called a " section corner.** Every sixth mile has a different
mark, and is called a " township comer.*'
* The surveyor should obtain and follow the instructions in force at the time the
original surveys were made in his locality.
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1/8 SURVEYING.
164. The Division into Townships. — From each "town-
ship corner " on any standard parallel auxiliary meridians are
run north to the next standard parallel. Since these meridians
converge somewhat toward the principal meridian, they will
not be quite six miles apart when they reach the next standard
parallel. But the full six-mile distances have been marked off
on this parallel from the principal meridian, and it is from
these township corners that the next auxiliary meridians will
start and run north to the next standard parallel, etc. Thus
each standard parallel becomes a " correction-line ** for the
meridians. The territory has now been divided into " ranges "
which are six miles wide, each range being numbered east and
west from the principal meridian. These ranges are then cut
by east and west lines joining the corresponding township
corners on the meridians, thus dividing the territory into
** townships,** each six miles square, neglecting the narrowing
effect of the convergence of the meridians. The townships
are numbered north and south from the " principal base-line.**
The fifth township north of this base-line, lying in the third
range west of the principal meridian, would be designated as
" town five north, range three west." Each township contains
thirty-six square miles, or 23,040 acres.
165. The Division into Sections. — ^The township is divided
into thirty-six sections, each one mile square and containing
640 acres. This is done by beginning on the south side of
each township and running meridian lines north from the
" section corners ** already set, marking every mile or " section
corner," and also every half-mile or *' quarter-section corner.**
When the fifth section corner is reached, a straight line is run
to the corresponding section corner on the next township line.
This will cause this bearing to be west of north on the west, and
east of north on the east, of the principal meridian. When this
northern township boundary is a standard or correction-line,
then the sectional meridians are run straight out to it, and thus
this line becomes a correction-Hne for the section-lines as well
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LAND SURVEYING, 179
as for the township-lines. The east and west division-lines
are run, connecting the corresponding section corners on the
meridian section lines, always marking the middle, or quarter-
section points. Evidently, to run a straight line between two
points not visible from each other, it is necessary first to run a
random or trial line, and to note the discrepancy at the second
point. From this the true bearing can be computed and the
course rerun, or the points on the first course can be set over
the proper distance. The sections are numbered as shown in
Figs. 51 and 52.
When account is taken of the convergence of meridians, the
sections in the northern tiers of each township will not be quite
one mile wide, east and west ; but as the section corners are set
at the full mile distance on the township-lines, the southern
sections in the next town north begin again a full mile in width.
In setting the section and quarter-section corners on the east
and west town lines the full distances are given from the east
toward the west across each township, leaving the deficiency
on the last quarter-section, or 40-chain distance, until the next
correction-line is reached, when the town meridians are again
adjusted to the full six-mile distances.
166. The Convergence of the Meridians is, in angular
amount,*
^= w sin ^ (Z + Z');
where m = meridian distance in degrees, or difference of longi-
tude, and L and L are the latitudes of the two positions. In
other words, the angular convergence of the meridians is the
difference in longitude into the sine of the mean latitude.
The convergence in chains of two township-lines six miles
apart, from one correction-line to another twenty-four miles
apart, in lat. 40"*, is
C = 24 X 80 X sin ^ ;
where r, in degrees, = ^ sin 40°, since one degree of longitude
in lat. 40*^ = 53 miles. Thus c = 4'.37 for each six-mile dis-
* From Eq. (G), Appendix D, when cos i J A is taken as unity.
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SURVEVmG.
tancc, east or west, in lat. 40°. Whence C = 2.42 chains,
which is what the northern tier of sections in the north range
between correction-lines lacks of being six miles east and west.
In a similar manner, we may find that the north sections in
Fig. 51.
79.40
80
80
80
80 1
80
6
5
4
3
2
I
79 92
79.92
79.92
79.92
79.92
79.92
7
8
9
10
II
12
79 94
79.94
18
17
16
15
14
1
>3 1
79-95
•
79-95
19
20
21
22
23
1
24
79.97
79-97
30
29
28
27
26
1
25
79.98
79.98
31
32
33
34
35
36
80
80 '
80
80
80
80
CORRECTION-LINE.
a town are about six feet narrower, east and west, than the
corresponding southern sections in the same town.
Figures 51 and 52 show the resulting dimensions of sections
in chains when no errors are made in the field-work. The
north and south distances are all full miles.*
In Fig. 51 it will be observed that in the northern tier of
sections the meridians must bear westerly somewhat so as to
meet the full-mile distance, laid off on the township-line.
* Of course all measurements in surveying are more or less inexact, and hence the
actual lengths on section lines de^te morc^ or less from these theoretical amounts.
LAND SURVEYING.
I8l
In Fig. 52 they continue straight north to the town line,
which is in this case a correction-line. If the distances on this
correction-line be summed they will be found to be 2.42 chains
short of six miles a^ above computed.
CORRECTION-LINE.
78.08
79.90
79.90
79.90
79.90
79 90
6
5
4
3
2
I
78.10
II
79.92
12
7
8
9
10
78.12
15
79-94
13
18
17
16
14
78.13
79-75
19
20
21
22
23
24
78. X4
19-91
30
29
28
27
26
25
78.16
79.98
3t
32
33
34
35
36
78.18
80
80
80 .
80
80
Fig. 52.
The law provides that all excesses or deficiencies, eithei
from erroneous measurements or bearings or from the conver-
gence of meridians, shall, so far as possible, be thrown into the
northern and western quarter-sections of the township.
167. Comer Monuments have been established on all
United States land surveys at all the corners of townships,
sections, quarter-sections, and meandered lines, except at the
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1 8^ SURVEYING.
point common to four-quarter sections, at the center of each
section. These monuments are of various composition, as :
{a) A stone with pits and earthen mound.
{6) '* *' " a mound of stone. *
(c) ** " ** bearing trees.
{d) A post in a mound of stone.
(e) '* " " ** " ** earth.
if) " ** with bearing trees.
(g) A mound without post or stone,
(A) A tree without bearing trees.
(/) '' ** with
Whenever possible, certain descriptive marks and letters are
cut on the stones, posts, or trees, such as described in the fol-
lowing sample field notes, taken from the Manual of Insiruc
iioTiSy issued by the United States Land Commissioner, Wash-
ington, D. C, in which full illustrations and descriptions are
found on all matters pertaining to the original surveys of
public lands. It should also be stated, that any of the styles
of marking above-named may be usjd for any kind of corner,
and that the styles described below are not limited to the pur-
poses there named.
STANDARD TOWNSHIP CORNER.
Set a — stone — x — x — x ins. — ins. in the ground for
Standard Cor. to Tps. 5 N., R*s 2 and 3 W., marked
Stone, wiUi f J t u j
MouS** S. C, with-6 notches on N., E., and W. edges, dug
pits 24 X 18 X 12 ins. crosswise on each line ; N., E.,
and W. of stone 6 ft. dist., and raised a mound of earth 2j ft.
high, 5 ft. base, alongside.
STANDARD SECTION CORNER.
Set a post 4 ft. long, 4 ins. square, 24 ins. in ground fo(
Standard Cor., to sees. 35 and 36, marked
^Sari'ng*' S. C. T. 5 N., R. 3 W., on N.;
Trees. *^ »
S. 36 on E., and
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LAND SURVEYING. 1 83
S. 35 on W. faces, with I notch on E. and 5 notches on W.
faces, from which
A — , — ins. diam. bears N. — <>,£. — Iks. dist., marked T.
5 N., R. 3 W. S. 36 B. T.
A — , — ins. diam. bears N. — °, W. — Iks. dist. marked T.
5 N., R. 3 W. S. 35 B. T.
A — , — ins. diam. bears S. — °, E. — Iks. dist., marked T.
5 N., R. 3 W. S. C. S. 35 and 36 B. T.
CORNER COMMON TO FOUR SECTIONS.
Deposited a marked stone (charred stake or quart of
charcoal) 12 ins. in the ground, for Cor. to Sees.
^iSstonc"* ^5' ^^' 25' ^^^ 3^' ^"S P^^^' 18x18x12 ins. in
each Sec, 5 J ft. dist., and raised a mound of earth
2 ft. high, 4^ ft. base over it.
In S. C. pit drove a stake 2 ins. square, 2 ft. long, 12 ins. in
ground, marked
T. 2 N., S. 25, on N. E.
R. 2 W., S. 36, on S. E.
S. 35, on S. W., and
S. 26, on N. W. faces with i notch on S. and E. edges^
QUARTER-SECTION CORNER.
Set a post 3 ft. long, 3 ins. square, with marked stone
Posrt in (charred stake or quart of charcoal), 12 ins. in the
ground, for \ Sec. Cor., marked \ S. on N. (or W.)
face; dug pits, 18 x 18 x 12 ins., N. and S. (or E. and W.) ft.
base of post ^\ ft. dist., and raised a mound of earth i^ ft.
high, 31^ around post.
168. The Subdivision of Sections.*— No interior section
lines were run by the United States Deputy Surveyors, but
* For full exposition of this method as given by U. S. General Land OflSce,
see Appendix I, p. 734- / r^r^rrl^
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1 84 SURVEYING,
quarter-section corners were set at the middle points of all the
four sides of all sections, except those in the north and west
tiers in each township. In order to satisfy the law, that so far
as possible all excesses and deficiencies should be thrown into
the north and west tiers of quarter-sections, the monuments
on these section lines are placed just 40 chains from the next
interior section lines. That is say, the east and west quarter-
section line in the north tier of sections lies 40 chains from and
parallel to the south side of such section, and the north and
south quarter-section line in the west tier of sections lies 40
chains from and parallel to the east side of such sections. In all
other cases the quarter-section lines are intended to be medial
lines. The location of the quarter-section corners, when set,
will control the position of these lines, however, so that nothing
remains to be done in making a resurvey but to run trial or
random lines through the section Jrom one quarter-section
corner to the opposite one, and by noting the errors, correct
one-half of them at the centre of the section, and so obtain the
point where the lines joining the opposite quarter-section comers
of a section intersect. This is the interior quarter-section corner.
In case no quarter-section corner has been or can be set on
one side of a section, the quarter-section line is to be extended
from the opposite corner by a true north and south or east and
west line.
The north and west tiers of quarter-sections in every town-
ship are called fractional quarters, and are divided again into
one full half-quarter and a fractional half-quarter. The north-
ern tier are so divided by an east and west line running just 20
chains north of the quarter-section line, and the western tier
of quarter-sections are divided by a north and south line lying
just 20 chains west of the quarter-section line, the N. W. quar-
ter of Section 6 being classed with the northern tier of quar-
ters. All other subdivided quarter-sections are divided into
half quarter-sections by medial lines run north and south, and
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LAND SURVEYING. 1 85
into quarter quarter-sections by medial lines run east and west
between the corresponding sides of the quarter-section.
169. To Run Out a Parallel of Latitude or a True East
and West Line. — A true east and west line is one which is at
every point at right angles to a meridian passing through that
point. It is therefore a constantly curving line, being always
deflected toward the north in the northern hemisphere. Any
line run on the earth's surface by prolongation by means of
any surveying instrument will be a great circle. If the mag-
netic needle always pointed directly north, a line run at right
angles to its direction would be a parallel of latitude. Since
the solar compass, or attachment, always orients itself in the
true meridian, any line run by it at right angles to the con-
stantly observed meridian will be a true east and west line.
This is the only instrument capable of running such a line
directly.
This method is not so accurate, however, as to use a transit,
and make frequent observations for azimuth. Then, starting
out on a true east and west line, run out a straight line by
prolongation (Art. 100, p. 95) for some twelve miles distance,
and make corrections northward for the points on the true
parallel. Then offset the proper distance, set the transit again
on the parallel, and either make a new observation for azimuth
or carry the old azimuth forward, correcting it to agree with
the new meridian. To do this two tables are required : one
to give the proper offsets from the great circle to fhe parallel
of latitude tangent to it at the initial meridian, and the other ,
to give the change in azimuth necessary to prolong the line
from a new meridian when no new observation for azimuth can
be obtained. These two tables* are combined in one on the
following page. The angles there given are measured from the
north point toward the point of tangency of the straight line
* Condensed from tables given in the ** Manual of Instructions," issued by the
Commissioner of the General Land Office, Washington, D. C, 189a
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i86
SURVEYING,
with the parallel, which is the initial point from which the
distances given in the table are measured. The convergence
of the meridians for the corresponding distances is 90°, minus
the angles given in this table.
The offsets are to be always measured to the north of the
great circle or tangent straight line, in the northern hemi-
sphere, and south from it in the southern hemisphere.
Having started from a given point due east, in latitude 40°,
for instance, and run out a straight line for six miles, we find
from the accompanying table that the true meridian is ob-
tained by turning off from east to south or from west to north
the angle 89^ 55' 38", and the true position of the parallel at
this point is 20.1 ft. north of the line. When twelve miles
have been run out in one continuous tangent line the angle
with the meridian is 89° 51' 17", and the parallel now lies 80.5
ft. north of the line.
ACUTE ANGLES WITH THE MERIDIAN. AND OFFSETS TO PARAL-
LELS, AT POINTS ONE MILE APART ON A GREAT CIRCLE OR
STRAIGHT LINE TANGENT TO THE PARALLEL AT THE INI-
TIAL POINT.
Latitude
I Mile from
Tangent Point.
.2 Miles.
3 Miles.
4 Miles.
■
AngH.
Offset.
Angle.
Oflfsct.
Angle.
Oflfset.
Angle.
Offset.
e
0 / /'
89 59 30
89 59 28
89 59 35
89 59 M
ft.
0.39
.4»
0 / //
89 58 50
89 58 AA
ft.
1.80
».94
0 / //
89 58 30
f9 58:.3
89 58 15
89 5807
ft.
0 / //
89 58 00
89 57 50
89 57 40
89 57 29
ft
6.17
6.67
7.20
7-75
38
49
89 59 '9
89 59 16
89 59 «3
1
89 58 39
a.o8
3.24
2.40
89 57 58
89 57 49
89 57 40
4.69
5.03
5.40
89 57 >8
89 «;7 06
89 5653
S-33
8.95
9.59
48
JO
89 59 10
89 5906
.74
0.79
89 58 90
?9 58 la
895805
89 57 56
a. 76
2.95
3x7
i9 57 30
89 57 »9
89 5707
89 5654
5. 79
6.ao
6.65
7.1a
89 56 25
89 56 09
89 55 53
10.39
11.04
It. 82
13.68
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LAND SURVEYING.
187
LatUnde
5 Miles.
6 Miles.
7 Miles.
8 MUc8.
Angle.
Offset.
Angle.
OflEsei.
Angle.
Offset.
Angle.
Offset.
0
30
32
^
38
40
42
ii
50
0 / /'
595730
89 57 >8
89 5705
895651
89 56 37
89 56 22
895606
89 55 49
89 55 3«
89 55 «3
89 54 SO
ft.
9.64
20.42
11.25
12.11
13.02
13.98
M-99
16.07
17.21
18.47
19.80
0 / //
89 5700
895645
895630
89 56 X3
8955 56
89 55 38
89 55 «9
89 54 59
89 54 37
8954 14
89 53 49
ft.
13.88
15.09
X6.90
J7-4I
X8.75
20. 11
21.59
33 '4
24.80
26.59
38.5a
0 / //
8956 30
89 56 12
8955 54
8955 36
8955 16
89 54 56
89 54 33
89 54 09
89 53 43
89 53 16
89 53 47
ft.
18.89
20.44
22.05
33.74
35.5a
27.40
39.38
3X.50
33.76
3S-19
38.82
0 1 II
89 5600
89 55 40
89 55 19
89 54 59
89 54 35
89 54 "
89 53 46
895318
895a 52
89 53 18
89 51 45
ft.
38.80
31 .ox
41.14
44. to
47.37
50.70
Latitude
9MUes.
10 MUes.
11 Miles.
12 MUes.
Angle.
Offset.
An^lc.
Offset.
Angle. Offset.
Angle.
Offset.
0
30
33
3I
38
40
42
I
50
0 / //
89 5530
89 5508
8^} 54 44
89 54 30
!9 53 55
89 «;3 38
89 52 59
8953 28
89 5« 56
89 51 20
89 50 43
ft.
36.4s
39.35
42.19
48.57
0 / //
89 55 00
89 54 35
89 54 09
89 53 43
89 53 X4
89 53 44
89 52 13
89 5t 38
89 51 02
89 5033
89 49 4«
•
ft.
38.55
41.71
45.00
48.45
53.08
55.91
59.97
73.86
79.23
0 / //
895430
89 5403
89 53 34
89 5304
89 53 33
89 52 00
8951 35
89 5048
89 50 08
89 49 25
89 48 39
ft.
46.65
50.47
r^
63.09
67.65
72.56
Z7-78
83.37
9 t II
89 54 00
89 53 30
89 52 TO
89 52 a6
89 51 53
59 51 17
89 5038
8949 58
8947 37
ft.
60.06
64.80
69.77
&??
86.35
93.57
106.36
1x4.08
FINDING THE AREA OR SUPERFICIAL CONTENTS OF LAND
WHEN THE LIMITING BOUNDARIES ARE GIVEN.
170. The Area of a Piece of Land is the area of the level
surface included within the vertical planes through the bound-
ary-lines. This area is found in acres, roods, and perches, or,
better, in acres only, the fractional part being expressed
decimally. Evidently the finding of such an area involves two
distinct operations, viz. : the Field-work, to determine the
positions, directions, and lengths of the boundary-lines ; and
the Computation, to find the area from the field-notes. There
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1 88
SURVEYING,
are several methods of making the field observations, giving
rise to corresponding methods of computation. Thus, the
area may be divided into triangles, and the lengths of the sides,
or the angles and one side, or the bases and altitudes measured,
and the several partial areas computed. Or the bearings and
distances of the outside boundary-lines maybe determined and
the included area computed directly. This is the common
method employed. Again, the rectangular coordinates of each
of the corners of the tract may be found in any manner with
reference to a chosen point which may or may not be a point
in the boundary, and the area computed from these coordi-
nates. These three methods will be described in detail.
I. Area by Triangular Subdivision.
171. By the Use of the Chain Alone. — In
ABCDEF be the corner bound-
aries of a tract of land, the sides
being straight lines. Measure
all the sides and also the diag-
onals AC^ AD, AE, and FB.
The area required is then the
sum of the areas of the four tri-
angles ABC, A CD, ADE, and
AEF. These partial areas are
computed by the formula
Fig. 53 let
Area = Vs{s — a){s — b){s — c\
Fig. S3.
where s is the half sum of the three sides a, b, c in each case.
For a Check, plot the work from the field-notes. Thus, take
any point as A and draw arcs of circles, with A as the com-
mon centre, with the radii AB, AC, AD, AE, and -^/^ taken to
the scale of the plot. From any point on the first arc, as B,
and with a radius equal to BC to scale, cut the next arc, whose
radius was AC, giving the point C. From C find D with the
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LAND SURVEYING, 1 89
measured distance CD^ etc., until F is reached. Measure FB
on the plot, and if this is equal to the measured length of this
line, taken to the scale of the drawing, the field-work and plot are
correct. It is evident the point A might have been taken any-
where inside the boundary-lines without changing the method.
172. By the Use of the Compass, or Transit, and Chain.
— If the compass had been set up at A the outer boundaries
could have been dispensed with, except the lines AB and AF,
All that would be necessary in this case would be the bear-
ings and distances to the several corners. We then have two
sides and the included angle of each triangle given when the
ar^'of each triangle is found by the formula:
Area = \ab sin C.
In this case there is no check on the chaining or bearings.
The taking-out of the angles from the given bearings could be
checked by summing them. This sum should be 360° when
A is inside the boundary-line, and 360° minus the exterior
angle FAB when A is on the boundary. If the boundary-
lines be measured also, then the area of each triangle can be
computed by both the above methods and a check obtained.
173. By the Use of the Transit and Stadia.*— Set up
at Ay or at any interior or boundary point from which all the
corners can be seen, and read the distances to these corners
and the horizontal angles subtended by them. The area is
then computed by the formula given in the previous article.
The distances may be checked by several independent read-
ings, and the angles by closing the horizon (sum -- 360°).
The above methods do not establish boundary-lines, which
is usually an essential requirement of every survey.
II. Area from Bearing and Length of the Boundary-lines,
174. The Common Method of finding land areas is by
means of a compass and chain. The bearings and lengths of
the boundary-lines arc found by following around the tract to
** The stadia methods are described in Chapter VIH.
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190 SURVEYING,
the point of beginning. If the boundary-lines are unobstructed
by fences, hedges, or the like, then the compass is set at the
corners, and the chaining done on line. If these lines are ob-
structed, then equal rectangular offsets are measured and the
bearings and lengths oi parallel lines are determined. In this
case the compass positions at any corner for the two courses
' meeting at that corner are not coincident, neither are the final
point of one course and the initial point of the next course,
the perpendicular offsets from the true corner overlapping on
angles less than 180° and separating on angles over 180''.
The chaining is to be done as described in Art. 4, p. 8, the
66-foot or Gunter's chain being used. Both the direct and the
reverse bearing of each course should be obtained for a check as
well as to determine the existence of any local attraction. For
the methods of handling and using the compass see Chapter II.
175. The Field-notes should be put on the left-hand page
and a sketch of the line and objects crossing it on the right-
hand page of the note-book. The following is a convenient
form for keeping the notes. They are the field-notes of the
survey which is plotted on p 192. It will be seen that the
"tree'* was sighted from each corner of the survey and its
bearing recorded. If these lines were plotted on the map
they would be found to intersect at one point. If the plot
had not closed, then these bearings would have been plotted
and they would not have intersected at one point, the first
line which deviated from the common point indicating that
the preceding course had been erroneously measured, either in
bearing or distance, or else plotted wrongly. In general such
bearings, taken to a common point, enable us to locate an
error either in the field-notes or in the plot. The bearings of
all division-fences were taken, as well as their point of inter-
section with the course, so that these interior lines could be
plotted and a map of the farm obtained. The " old mill " is
located by bearings taken from comers B and G* The reverse-
bearings are given in parenthesis.
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LAND SURVEYING.
591
FIELD NOTES— COMPASS SURVEY.
Oct. 23. 1885.
No. of
Course.
Point.
Bearing.
Disunce
along the
Course.
Remarks.
I
Bearing-lree
Pasture Fence
Yard *•
Orchard ** *.!!!!
Comer B
S. 76' 50' E.. . .
West
Ch.
7.20
9-75
11.54
13.90
25.42
True bearings given.
Variation of needle 5^
50' east.
Henry Flagg,
Compassman.
Courses i and 2 are
along the centres of
the highway.
«t
4(
Wt.= i
South
(North)
B.T
N. 54'*I5'E...
N. 58* E. . .
North
12.50
24.10
34-68
■
Old Mill
2
Wt.= i
Fence
Corner C
S. 89*»55'E....
(West)
3
Wt.= 3
B. T
N. 22° 20' W..
N. 26' 45' W.. .
N. 6i''45' W...
9.90
10.70
12.45
24.00
Old Mill
Fence
Mill Creek
Fence
N. 64" W
N. 27^ 40' E. . .
(S. 27^ 45' W.)
Corner D
4
Wt.= 2
B. T
S. 85* W
N. 19° 10' W..
(S. 19'' 15' E.)..
7.40
Corner E
5
Wt.= 2
B.T
S. 62'*30' W...
South
15.80
25.58
Fence
Corner F
N. 86° 50' W..
(S. 86^ 45' E.)..
6
Wt.= 5
B. T
S. 40** 15' E....
0.30
0.80
1.50
N bank Mill Creek.
s. •* •*
Corner G
s. 47^30' w...
(N.47°3o'E.).
7
Wi.= 3
Fence
S. 32'' E
0.00
0.00
3.00
6.00
9.00
12.00
13.60
13.60
Offset, 0.40
.60
.80
* * .70
* • . ao
** .20
Corner H
S. 77'45' W...
(N. 77^ 45' E.).
8
Wt.= i
Corner A
S. 89° W
(N. 89'' E.)....
3.53
^:^ V-r^
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192
SURVEYING.
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LAND SURVEYING,
193
Fig. 55.
COMPUTING THE AREA.
176. The Method stated. — In
Fig- 5S>* let ABCDE be the tract whose
area is desired. Let us suppose the
bearings and lengths of the several
courses have been observed. Pass a
meridian through the most westerly
corner, which in this case is the corner
A, Let fall perpendiculars upon this
meridian from the several corners, and
to those lines drop other perpendicu-
lars from the adjacent corners, as shown
in the figure. Then we have:
Area ABCDE = bBCDfb - bBAEDfb
= bBCe + eCDf - iJ)BA + AEa + aEDf). (i)
Hence twice the area ABCDE is
2 A = {bB + eC)Bc + {eC-{-fD)Dd
- {bB)Ab - {aE)Aa — {aE -\'fD)Eg.
(2)
We will now proceed to show that these distances are all
readily obtained from the lengths and bearings of the courses.
177. Latitudes, Departures, and Meridian Distances.—
Tlie latitude of a course is the length of the orthographic pro-
jection of that course on the meridian, or it is the length of the
course into the cosine of its bearing. If the forward bearing
of the course is northward its latitude is called its northing, and
is reckoned positively ; while if the course bears southward its
latitude is called its southing, and is reckoned negatively.
♦ The lines OD and OX in this figure are used in art. iqi.
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194 SURVEYING,
Tin departure of a course is the length of its orthographic
projection on an east and west line, or it is the length of the
course into the sine of its bearing. If the forward bearing of
the course is eastward its departure is called its easting, and is
reckoned positively ; while if its forward bearing is westward
its departure is called its westing, and is reckoned negatively.
The meridian distance of a point is its perpendicular dis-
tance from the reference meridian, which is here taken through
the most westerly point of the survey.
The meridian distance of a course is the meridian distance
of the middle point of that course ; therefore
The double meridian distance of a course is equal to the sum
of the meridian distances to the extremities of that course.
The D. M. D.*s of the two courses adjacent to the reference
meridian are evidently equal to their respective departures.
The D. M. D. of any other course is equal to the D. M. D. of
the preceding course plus the departure of that course plus
the departure of the course itself, easterly departures being
counted positively and westerly departures negatively. This
is evident from Fig. 55.
Thus in Fig. 55 Dd is the latitude and dC\s the departure
of the course DC, If the survey was made with the tract on
the left hand, then the latitude of this course is positive and
the departure negative ; while the reverse holds true if the
survey was made with the tract on the right hand. In this
discussion it will be assumed that the survey is made by going
around to the left, or by keeping the tract on the left hand,
although this is not essential. The D. M. D. of this course
CD \sfD + eC\ or it is the D. M. D. of BC+ cC-^{ — dC),
In equation (2), art. 176, the quantities enclosed in paren-
theses are the double meridian distances of the several courses,
all of which are positive, while the distances into which these
are multiplied are the latitudes of the corresponding courses.
If we go around towards the left the latitudes of the courses
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LAND SURVEYING, 195
ABy DE, and EA are negative, and therefore the correspond-
ing products are negative, while the latitudes of the courses
BC and CD being positive, their products are positive.
We may therefore say that twice the area of the figure is
equal to the algebraic sum of the products of the double meridian
distances of the several courses into the corresponding latitudes^
north latitudes being reckoned positively and south latitudes
negatively, and the tract being kept on the left in making the
survey. If the tract be kept on the right in the survey, then
the numerical value of the result is the same, but it comes out
with a negative sign.
178. Computing the Latitudes and Departures of the
Courses. — Since the departure of a course is its length into
the sine, and its latitude its length into the cosine, of its bear-
ing, these may be computed at once from a table of natural or
logarithmic sines and cosines. When bearings were (formerly)
read only to the nearest 15 minutes of arc, tables were used
giving the latitude and departure for all bearings expressed in
degrees and quarters for all distances from i to 100. Such
tables are called traverse tables. It is customary now, how-
ever, to read even the needle-compass closer than the nearest
15 minutes; and if forward and back readings are taken on all
courses, and the mean used, these means will seldom be given
in even quarters of a degree. If the transit or solar compass is
used, the bearing is read to the nearest minute. The old style
of traverse table is therefore of little use in modern survey-
ing. The ordinary five- or six-place logarithmic tables of
sines and cosines are computed for each minute of arc, and
these may be used, but they are unnecessarily accurate for or-
dinary land-surveying. For this purpose a four-place table is
sufficient. If the average error of the field-work is as much as
I in 1000 (and it is usually more than this), then an accuracy
ot I in 5000 in the reduction is evidently all-sufficient, and this
is about the average maximum error in a four-place table; that
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196
SURVEYING.
is, the average of the maxiirum errors that can be made in the
different parts of the table.
Table III. is a four-place table of logarithms of numbers
from I to 10,000, and Table IV. is a similar table of logarithms
of sines and cosines, fromo to 360 degrees. If a transit is
used in making the survey, and if it is graduated continu-
ously from o to 360 degrees, then the azimuths of the several
sides are found, all referred to the true meridian or to the first
side. If it is desired now to take out the latitudes and de-
partures, the same as for a compass-survey, where the bearings
N
18d'
W90
y^
~~~^^
/ '"^
L-h \
/ D —
D-h \
I ■•"
J
\ ^""
D+ /
370E
Fig. 56.
of the sides are given directly referred to the north and south
points, it may be done by Table IV.
Since the log sine changes very fast near zero and the log
cosine very fast near 90°, the table is made out for every min-
ute for the first three degrees from these points ; for the rest
of the quadrant it gives values 10 minutes apart, but with a
tabular difference for each minute. It is very desirable to
make the table cover as few pages as possible for convenience
and rapidity in computation. In this tabic the zero-point is
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LA!^D SURVEYING.
197
south and angles increase in the direction SW/NE, so that in
the first quadrant both latitudes and departures are negative.
In the second quadrant latitude is positive and departure nega-
tive, in the third both are positive, and in the fourth latitude
is negative and departure positive. These relations are shown
in Fig. 56. For any angle, falling in any quadrant, if reckoned
from the south point in the direction here shown, the log sin
(for departure) and log cosine (for latitude) may be at once
found from Table IV. If these logarithms are both taken out
at the same time and then the logarithms of the distance from
Tablelll., this can be applied to both log sin and log cos, thus
giving the log departure and log latitude, when from Table III.
again we may obtain the lat. and dep. of this course, giving
these their signs according to the quadrant in which the azi-
muth of the line falls.
If Table IV, is to be used for bearings of lines as given by a
needle-compass, then enter the table tor the given bearing, in
the first set of angles, beginning at o and ending at 90°.
Example: Compute the latitudes anv! departures of the survey plotted in
Fig- 55* P- 193. by Tables III. and IV. The following are the field-notes as ihey
would appear, first, as read by a transit and referred to the true meridian; and,
second, as read by a needle compass:
Sutlon.
Azimuth referred to
the South Point.
Compass bearing.
Distance.
A
290" 45'
S. 69'' 15' E.
7.06
B
zi?** 15'
N. 37" 15' E.
5-93
C
140" 30'
N. 39' 30' W.
6.00
D
57" 45'
S. 57" 45' W.
4.65
E
30** 00'
S. 30" 00' w.
4.98
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198
SURVEYING,
The following is a convenient form for computing the lati-
tudes and departures :
Course
AB
4th Q.
Course
BC
3d a
Course
CD
adQ.
Course
DB
isiQ.
Course
EA
181 Q.
9.9708
9.7820
9-8035
9.9272
9.6990
.8488
ni-h^
.7782
.6675
.6972
.8196 1 .5551
.5817
•5947
.3962
+ 6.60 +3-59
— 3.82
-3.93
-2.49
9-5494
9.9009
9.8874
9.7272
9-9375
.848S
.7731
.7782
.6675
.6972
.3982
.6740
.6656
.3947
-6347
— 2.50
+ 4.72
+ 4.63
— 2.48
-4-31
log sin (dep.) =
log dist. =
log dep. =
Departure =
log cos (lat.) =
log dist. =
log lat. = «
Latitude =
It is seen that Table IV. answers equally well for either set
of bearings,and also that Table III. would have given the lati-
tudes and departures to the fourth significant figure as well as
to the third. If the proper quadrant is given for each course
in the heading as .shown above, then the signs may be at once
given to the corresponding latitudes and departures.
179. Balancing the Survey. — If the bearings and lengths
of all the courses had been accurately* determined, the survey
would "close;" that is, when the courses are plotted succes-
sively to any scale the end of the last course would coincide
on the plot with the beginning of the first one. Furthermore,
the sum of the northings (plus latitudes) would exactly equal
^he sum of the southings (minus latitudes), and the sum of the
* The error of closure simply shows a want 0/ uniformity of measurement,
for if all the sides were in error by the same relative amount, the survey would
close just the same. For instance, if an erroneous length of chain were used,
the survey might close but the area be considerably in error. See Arts. 1 80
and 182.
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LAND SURVEYING,
199
castings (plus departures) would exactly equal the sum of the
westings (minus departures). It is evident that such exactness
is not attainable in practice, and that neither* the north and
south latitudes nor the east and west departures will exactly
balance, there always being a small residual in each case.
These residuals are called the errors of latitude and departure
respectively. The distribution of these errors is called bal-
ancing the survey.
In the form for reduction of the field-notes given below,
wherein this example is solved, it is seen that the error of lati-
tude is 6 links and the error of departure is 5 links. The dis-
tribution of these errors is made by one of the following :
FORM FOR COMPUTING AREAS FROM BEARINGS AND DISTANCES
OF THE SIDES.
Sta.
Courses.
Dif. Lai.
Departure.
Balanced.
Q
4-
Area.
tioos.
BeariogB.
Dist.
N.
■f
S.
£.
+
W.
Lat.
-3 52
Dcp.
Area.
A
S.69«is'E.
Ch.
7.06
3.50
6.60
+ 6.61
6.61
16.66
B
N. y,- 15' E.
5-93
4.7a
....
3 59
....
+ 4.71
+ 3.60
16.8s
79.23
....
C
N. 39* 30' W.
6.00
4.63
3.83
+ 463
-3.81^
x6.6i \ 76.74
D
S.57'*45'W.
465
348
3-93
-3.49
— 3 9a
8.88
....
33.11
B
S. 3o»oo'W.
4.98
....
4-3«
3.49
-4.32
-3.48
3.48
....
10.71
38.63
9-35
9.39
10.19
10.34
«55-96
49.48
9.29
10. 19
49.48
Erro
rinlat
= .06
Error
in dep
= .05
«^
106.48
8863
= 110366.
Area = 53.24 sq.ch.
= 5.334^^/f
ir
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200 ^ SURVEYING,
RULES FOR BALANCING A SURVEY.
Rule i. As the sum of all the distances is to each particular
distance, so is the whole error in latitude {or departure) to the cor-
rection of the corresponding latitude (or departure), each correc-
tion being so applied as to diminish the whole error in each
case.*
Rule 2. Determine the relative difficulties to accurate
measurement and alignment of the several courses, selecting
one course as the standard of reference. Thus, if the standard
course would probably give rise to an error of i, determine
what the errors for an equal distance on the other courses
would probably be, as 1^,2, 1,0.5 ^^c. Multiply the length
of each course by its number, or weight, as thus obtained.
Then we would have :
As the sum of all the multiplied lengths is to each multiplied
leftgth, so is the whole error in latitude {or departure) to the cor-
rection of the corresponding latitude {or departure), each correc-
tion being so applied as to diminish the whole error in each
case.
These two rules are based on the assumption that the error
of closure is as much due to erroneous bearings as to erroneous
chaining, which experience shows to be true in needle-compass
work.
If, however, the bearings are all taken from a solar compass
(or attachment) in good adjustment, or if the exterior lines are
run as a traverse with a transit, so that the angles of the pe-
rimeter are accurately measured, then the above assumption
does not hold, as it is highly probable that the error of closure
is almost wholly due to erroneous chaining. Especially would
this be highly probable if the azimuth is checked by occupying
♦ For finding the corresponding corrections to the lengths of the courses
themselves, see note under tabular computation, p. 204. This was first sug-
gested by Mr. Antonio Llano in Eng. News, Nov. 23, 1899.
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LAND PURVEYING. 20I
the first station on closing and redetermining the azimuth of the
first course, as found from the traverse, and comparing it with
the initial (true or assumed) azimuth of this course. If it thus
appears that the traverse is practically correct as to angular
measurements, it may be fairly assumed that the error of
closure is almost wholly due to erroneous chaining. In this
case use
Rule 3. As the arithmetical sum of all the latitudes is to any
one latitude, so is the whole error in latitude to the correction to
the corresponding latitude, each correction being so applied as
to diminish the whole error in each case. Proceed similarly
with the departures.*
In the solution given on p. 199 the first rule is applied. In
ordinary farm-surveying it is not common to give the lengths
of the courses nearer than the nearest even Hnk or hundredth
of a chain. In balancing, therefore, the same rule may be
observed.
180. The Error of Closure is the ratio to the whole pe-
rimeter of the lengtli of the line joining the initial and final
points, as found from the field-notes. The length of this line
is the hypotenuse of a right triangle of which the errors in
latitude and departure are the two sides. Its length is there-
fore equal to the square root of the sum of the squares of
these two errors. This divided by the whole perimeter gives
the error of closure, which ratio is usually expressed by a
vulgar fraction whose numerator is one, being ^fr ^^ ^^^
above example.
The error of closure for ordinary rolling country should not
* It is evident that the courses could here be weighted for different degrees
of difficulty in the chaining; but instead of multiplying the lengths of the
courses by their weights, multiply the latitudes and departures by the weights
of the corresponding courses, and then distribute the errors in latitude and
departure by these multiplied latitudes and departures.
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202 SURVEYING.
be more than i in 500. In city work it should be less than I
in 1000, and should average less than i in 5000.
181. The Form of Reduction. — On p. 199, the ordinary
form of reduction is shown. Here the courses are not weighted
for different degrees of difficulty in chaining; and since it
was a compass-survey the effect of erroneous bearings is sup-
posed to equal that from erroneous chaining, and so the first
rule for balancing is used. The balanced latitudes and de-
partures having been found, the double meridian distances are
next taken out. In taking out these it is preferable to begin
with the most westerly corner^ whether this be the first course
recorded or not. In the example solved on p. 199, it is the
first corner occupied, but in that given on p. 206 it is not the
first course. By beginning with the most westerly corner
(which is equivalent to passing the reference meridian through
that corner), all the double meridian distances will be positive ;
otherwise some of them may be negative. If attention be
paid to signs we may begin at any corner to compute the
double meridian distances.
A check on the computation of the D. M. D.'s is that, when
computed continuously in either direction and from any cor-
ner, the numerical value of the D. M. D. of the last course
must equal its departure. This is a very important check and
must not be neglected, as it proves the accuracy of all the D.
M. D.'s.
We are now able to compute the double-areas according to
equation (2), art. 176, since the terms entering in that equation
have their numerical values determined. The several products,
being the partial double-areas, are written in the last two col
umns, careful attention being paid to the signs of these prod-
ucts. Thus, when the reference meridian is taken through the
most westerly corner, then all the D. M. D/s are positive and
the results take the sign of the corresponding latitude. If
some of the D. M. D.'s are negative, then the signs of these par
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LAND SURVEYING. ^0%
tial areas are opposite to those of the corresponding latitudes.
The algebraic sum of the partial double-areas is twice the
area of the figure, as shown in eq. (2), Art. 176. If the dis-
tances are given in chains, then the area is given in sq.
chains, and dividing by ten gives the area in acres. If the dis-
tances were given in feet, as it often is, being measured by a
100-foot chain or tape, then the area is in sq. feet, and this
must be divided by 43560, the number of sq. feet in one acre,
to give the area in acres. This is best done by logarithms, as
shown in the example solved on p. 206. It is preferable to ex-
press areas in acres and decimals rather than in roods and
perches, as was formerly the custom.
On the following page is the reduction of the field-notes
given on p. 191. Here the several courses have been weighted
for various degrees of difficulty in the chaining. Thus, the first
and second courses were along the public highway and on even
ground. These are taken as the standard and given the
weight unity. The third course is on very uneven ground and
is judged to give rise to about three times the error of courses
one and two per unit's distance. It is therefore weighted
three. The proper weight to give to the several courses is
thus seen to depend on the character of the obstructions to ac-
curate work, and represents simply the judgment of the sur-
veyor as to the probable relation of these sources of error.
The short course FG was very difficult to measure, as there
were precipitous bluffs, and the course GH was also on very
uneven ground.
Following the column of weights in the tabular reduction
are the multiplied distances ; the errors of latitude and depart-
ure are distributed according to the results in this column by
Rule Two, p. 200. This survey was also made with a needle-
compass.
In the following example the transit was used, and the
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204
SURVEYING,
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LAND SURi^EYI^G,
205
survey began at A, The azimuth of the line AB (Fig. 57)
was found by a solar attachment,
and then the other courses ran as
a traverse, the horizontal limb of
the transit being oriented by the
back azimuth of the last course.
The azimuths of the courses are
all referred to the south point as
zero, and increase in the direction
SWNE. After the last course
FA was run, the instrument was
carried to A and oriented by a
back sight on /^and the azimuth
of AB again determined. This
agreed so well with the original
azimuth of this course that the
azimuths of all the courses were
proved to be correct.f
The error of closure is therefore due to the chaining alone.
A hundred-foot chain was used so that the distances are all
given in feet. The obstructions to chaining were about uni-
form, so the courses are all given equal weight. In balancing.
Rule Three must be used, since the errors are supposed to
come only from the chaining.
If the errors in latitude and departure had been distributed
by Rule One, or in proportion to the lengths of the courses,
the resulting area would have been 56.41 acres, a difference of
0.07 acres, or about one eight-hundredth of the total area.
182. Area Correction due to Erroneous Length of
Fig. s7.*
* The lines MB and 0<y in this figure are used in art. 192.
t From the azimuth check here obtained, as compared to the errors in lat-
itude and departure, decide whether the latter are due mostly to the chaining
or whether the errors in azimuth have had an equal influence, and so determine
whether to use rule i or rule 3 in balancing.
14
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2o6
SURVEYING.
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LAND SURVEYING, 207
Chain. — If the measuring unit has not the length assigned to
it in the computation, then the computed area will be errone-
ous. Such an error will not show in the balancing of the work
or elsewhere, and hence an independent correction must be ap-
plied for this error. If the chain was too long by one one-
thousandth part of its length, for instance, then all the courses
are too short in the same ratio. And since similar plane fig-
ures are to each other as the squares of their like parts, we
would have
true area : computed area :: (looi)' : (looo)*,
or true area = \%%% computed area (nearly) ;*
or, in general, if / = length of chain and Al = error in length,
being positive for chain long and negative for chain short, and
if Al is small as compared with /, as it always is in this case,
then if we let
A = true area. A' = computed area
Cj^ = correction to computed area,
and A = relative error of chain,
we have
jl ^ i-±^A' = (I + 2A)A';
whence, A — A^ = Cji= 2AA\
That is to say, the relative area correction due to erroneous
length of chain is twice the relative error of the chain, being
positive for chain long, and negative for chain short.
•The error in this approximation is one one-millionth in this case, and
would always be inconsiderable in this class of problems.
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208 SURVEYING,
FINDING THE AREA OR SUPERFICIAL CONTENTS OF LAND
WHEN THE RECTANGULAR COORDINATES OF THE COR-
NERS ARE GIVEN WITH RESPECT TO ANY POINT AS AN
ORIGIN.
183. Conditions of Application of this Method.—
Where many tracts of land, all bounded by straight lines, art
somewhat confusedly intermingled, as is the case in many of
the older States, and where the area of each tract over an ex-
tended territory is to be found, this method is greatly to be
preferred to that by means of the boundary-lines. In this case
it is only necessary to make a general coordinate survey of the
whole territory, as described in Chapter VIIL, on Topographi-
cal Surveying, using the stadia for obtaining distances, and be-
ing careful to locate every corner of each tract. If areas alone
are required, no attention need be paid to the obtaining of
elevations for contour lines, and so the work is greatly facilitated.
A transit and two or three stadia rods would be the instru-
ments used. The survey would then be carefully plotted and
the coordinates measured on the sheet, or they could be com-
puted from the field-notes. If the plotting is carefully done
the former method is preferable. It is best to choose the
origin of coordinates entirely outside the tract and so that the
whole area falls in one quadrant, thus making all the codr-
dinates of one sign.
Large tracts of mineral land are sometimes acquired by
large companies, including perhaps hundreds of individual es-
tates. In such cases a topographical map of the region is
necessary; and when this survey is made, a little extra care to
obtain all the " corners'* of private claims will enable the areas
of all such lots to be determined with great accuracy and at
small additional cost. The method probably has no advaa
tages when the area of but a single tract is desired.
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LAND SURVEYING.
2og
184. The Method of Finding the Area from the Rec-
tangrular Coordinates of the Comers is as follows :
Let Fig. 58 be the same tract as that given in Fig. 55, and
Ve
Vc
D
V^^^^ \
.^_.\
C
"'^^^
Pig. 58.
let the origin be one chain west olA and three chains south of
B. Then, from the balanced latitudes and departures for this
case, given on p. 199, we find the following coordinates of the
corners j'a, j'ft, €;tc., denoting the latitudes of the corners Ay B^
etc., and similarly with Xa, Xf,, etc., for departures:
n = 5-52, ^* = 3-00, jfa ^ 7'7h J'a = 12.33, ye = 9-84*
ir^ = 1.00, X(, = 7.61, x^ = 1 1.2 1, Xa = 7.40, x^ = 3.48.
The area of the figure ABCDE is equal to the areas
y^BCy, +ycCDy^ - \y^Dy^ +y,A£y,+y^Ay,] ;♦
or
<4 = K( J'c -yb) (^ft +j^c) + {ya - yc) {^c + ^d) - <jd-y^ (^d+^*)
♦ Here y^^ y^^ etc., are used to designate point? and not ordinatcs. In the
foUowing equations they are ordinates.
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2IO SURVEYING.
By developing equation (i) we obtain
-4 = i {.ya^t-yaXh +yb^a —y^Xc +y^b —y<^d
+ydXc - ya^e +yeXd - ye^a]* (2)
From this we may obtain either of the following :
+yd (^o — Xe) +ye i^d - ^a)] ;
or
+ Xd{yc'-ye)+Xe{yd-ya)l .
From these equations we may obtain the following
(3)
RULE FOR FINDING THE AREA OF A CLOSED FIGURE
BOUNDED BY STRAIGHT LINES FROM THE RECTANGULAR
COORDINATES OF THE CORNERS.
Multiply tlie \ ^ji„^f^ c lo each corner by the difference be-
tween the \ ^ A c/riccu^ i ^f ^^^ ^^^ adjacent corner s, always making
the subtraction in the same direction around the figure^ and take
half the sum of tJte products.
The student will observe that this is simply a more general
case of the former method of computing the area from the
latitudes and double-meridian distances.
* If these co5r(li nates be arranged thus :
then in accordance with formula (2), the area is equal to the sum of the products
of the quantities joined by the broken Unes minus the sum of the products of the
quantities joined by the full lines.
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LAND SURVEYING,
2Ii
185. The Form of Reduction for this case is given below.
Corner.
Ordi nates
Abscissae
(-r).
Difference
between Alter-
nate Abscissse.
Double Areas.
A
5.52
1. 00
- 4- 13
— 22.80
B
3.00
7.61
— 10.21
— 30.63
C
7.71
II. 21
+ .21
+ 1.62
D
12.33
7.40
+ 7.73
+ 95.31
£
9.84
3.48
+ 6.40
+ 62.98
Plus areas = 159.91
Minus areas = 53.43
2 ) 106.48
Area = 53 • 24 sq. chns.
= 5 . 324 acres.
This is the same result as found on p. 199 by the other
method, as it should be, since the same balanced latitudes and
departures were used in each case.
It is also evident that after the balanced latitujles and
departures are obtained for the ordinary perimeter-survey, the
area may be computed by this form — from equations (3), p.
210, if preferred. Or, if the coordinates of the corners are
taken at once from a map, or computed from traverse lines,
the bearings and lengths of the courses joining such corners
could readily be computed. Thus, the length of any course,
■^5)', while its bearing is
as BC, is BC = nxT-W^VUo
the arc whose tan is -^ ^.
186. Supplying Missing or Erroneous Data.—In any
closed survey there are two geometric conditions that must
be fulfilled, viz. :
1. The sum of all the latitudes must be zero.
2. The sum of all the departures must be zero.
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ili ^VRVEyiNG.
These two conditions give rise to two corresponding equa-
tions.
If /„ /„ /„ etc., be the lengths of the several courses, and if
^i» ^«» ^»» etc., be their compass-bearings, then our two geo-
metric conditions give
/, sin ^1 + A sin ^, + A sin ^, -fetC-> =0- • • (0
/, cos «?, + /, cos B^ + /, cos B^ + etc., = o. . . (2)
Since we have two independent equations, we can solve for
two unknown quantities. These two unknowns may be any
two of the functions entering in the above equations. Thus,
if any two distances, any two bearings, or any one distance
and any one bearing are missing, they may be found from
these equations. Or, if but one bearing or distance is missing,
it may be found from one of these equations and the other
equation used for balancing either the latitudes or departures.
When all bearings and distances are given, these equations are
really used in balancing ; but if they are both used to deter-
mine missing quantities, there can be no balancing of errors,
for when the missing quantities are computed by these equa-
tions, both latitudes and departures will exactly balance. In
other words, all the errors of the survey are thus thrown into
the^e two quantities.
This artifice should therefore never be resorted to except
where it is impracticable to actually measure the quantities
themselves in the field.
There are four cases to be solved :
I. Where the bearing and length of one course are un-
known.
II. Where the bearing of one course and length of another
are unknown.
. III. Where two bearings are unknown.
IV. Where two lengths are unknown.
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LAND SURVEYING. 2\l
The bearings will be reckoned from both north and south
points around to the east and west points, as is common in
compass surveying. Then the length of a course into the sin
of its bearing gives its departure, and into the cos of its bear-
ing gives its latitude. North latitude is plus and south latitude
minus; east departure plus and west departure minus.
In every case let the sum of the departures of all known
courses, taken with the opposite sign, be Z>, and the sum of
their latitudes, taken with the opposite sign, be Z. Then D
and L are the departure and latitude necessary to close the
survey.
Case I. — Bearing and length of one course unknown.
The two condition equations here become
4 sin e^z=iD\
/«. cos ft
L = z. } <3)
D
Whence tan tf ^ = y- (4)
Having found the bearing, find /^ from either of equations
(3). Particular attention must always be paid to the signs of
D and L, Evidently sin 6^ (dep.) and cos 6^ (lat.) have the
same signs as D and L respectively, whence the quadrant
which includes the bearing may be determined and the proper
letters applied. For this purpose Fig. 56 may be consulted.
Case II. — The bearing of one course and the length of another
unknown.
In this case let a be the known bearing of the course whose
length is unknown, and let / be the known length of the course
whose bearing is unknown. Then we have
4 sin ^ + /sin ^„ = Z?; |
4cos^ + /cos<?„ = Z. ) ^^^
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314 SURVEYING.
If we let sin a^^s, and cos a^c^we have
4 = jZ>+r2:± vr -(/?• + L') + {sD+cLy. . (6)
Here there are two values of /^ which will satisfy the equa-
tion, and so there are two solutions to the problem. If the
surveyor has no knowledge whatever of either the unknown
length or bearing, the problem is indeterminate. If he has
seen the tract he could usually tell which lengtli or which
resulting bearing was the correct one, when the problem would
become determinate. When /^ is found, substitute in one of
equations (5) and find ff^. Pay careful attention to the signs
of the trigonometrical functions of all bearings. When the
two unknown courses are nearly at right angles with each
other the problem is impracticable.
Case III. — IVAen two bearings are unknown.
Let /' and /" be the known lengths of the courses whose
bearings are unknown. Then the equations become
/' sin ^^ + /'' sin e^^D\
/'cos6^^+rcos
:::.,1 <')
Whence cos p, = £f -[- L* J • • • W
Where A = ^777 "•
This case is also indeterminate unless one is able to tell
which of the two sets of bearings is the correct one.*
Case IV. — IVAen the lengths of two courses are unknown.
* And if the unknown sides are parallel, the problem is indeterminate.
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LAND SURVEYING. 21 5
Let a and b be the known bearings of the courses whose
lengfths are unknown.
Our equations here become
4 sin ^? + 4 sin ^ = /?;!
4cos^i + 4cos* = Z. f ^^^
whence 4= . Zs»n^~Z>cos^ >
sm ^? cos ^ — cos ^? sm * ) ^ ^
This case is determinate, except when the unknown sides
are parallel.
In case there is but one unknown, then either one of equa-
tions (3) will solve. In taking out either the sine or the cosine
from the tables, however, two angles will always be found
equidistant from the east or west point if the sine, and equi-
distant from either the north or south point if the cosine,
either of which may be chosen. In such case both sine atid
cosine must be found, when the signs alone of these two func-
tions will determine the quadrant in which the bearing is found.
Hence, if the single unknown is a bearing ; both of the equa-
tions (3) must be used in order to determine which of the two
bearings given by the table is the correct one, but one alone is
sufficient to obtain the numerical value of the bearing. Thus,
if the sine equation is used to compute the bearing, then the
latitude may be taken out for the given length and bearing ;
and these will then not balance, but will have to be balanced
in the usual way, while the departures will, of course, balance,
since tlie residual departure D necessary to close the survey as
to departures was used to compute the corresponding bearing.
The reverse of this would be true if the cosine equation were
used to compute the bearing.
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il6 SURVEYING,
PLOTTING THE FIELD-NOTES.
187. To plot a Compass Survey select a point for the
initial station, and pass a meridian through it in pencil. By
means of a semicircular protractor, such as is shown in Fig.
44, mark the bearing and draw an indefinite line from the sta-
tion point. On this line lay off to scale the length of the
course, thus establishing the next corner. Through this draw
another pencil meridian, and proceed as before. If the plot-
ting is perfect the length of the line joining the final with the
initial point, taken to scale, is the error of closure of the sur-
vey; and the horizontal and vertical components of this line,
taken to scale, should be the errors in departure and latitude
respectively as obtained by the computation.
If preferred, the bearings of the successive courses may be
so combined as to give the deflection-angle at each station, and
these laid off from the preceding course as already drawn.
Errors are more likely to accumulate in the plot by this
method, however, than by that first given.
Again, the rectangular co6rdinates of the several corners
may be computed and these plotted from a pair of rectangular
axes, but this is not a common practice.
For the plotting of transit surveys, especially where the
stadia is used, see Chapter VIII.
THE AREAS OF FIGURES BOUNDED BY CURVED OR IRREGULAR
LINES.
188. The Method by Offsets at Irregfular Intervals.
— Where a tract of land is bounded by a body of water, as a
stream or lake, it is customary to run straight lines as near the
boundary as practicable and then to take rectangular offsets
at selected intervals from these bordering-lines to the irregular
boundary. These small areas are then computed as trapezoids,
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LAND SURVEYING.
217
sjM s.'IS
Fig. 59.
the distance along the base-line being the altitude and the half-
sum of the adjacent offsets being the mean width. The oflfsets
should therefore be run at such intervals as to make this
method of computation sufficiently accurate. Such offsets
were taken from the course GH in Fig, 54, the notes for which
are given on p. 191.
The work of computation may be shortened by using a
modified form of the ipethod
of areas from the rectangular
coordinates of the corners,
which, in this case, are the ends
of the offset lines. Let Fig,
59 be an area to be determined
from the offsets from the line
AK. The position and lengh of the offsets are given. Take
the origin at A and let the distances along AK be the abscissae,
and the lengths of the offsets be the ordinates. Using the
second of equations (3), p. 210, we have
A = *i W j» --y^ + x^{ya -yc) + ^c(n -yd)
+ Mfe-ya) + ^gij^f -yh)
+^h{yg-'yk)+^k{yh'-ya)']' (0
But here x^, x^, ya, and y^^ are all zero ; also Xj^ = Xj,, hence
this equation becomes
A = il^c{yb- yd) + ^d{yc- ye) + ^e(yd- yh
+Mye-yg)+^g {yf-yh)+ ^h{yg+yh)l{2)
From eq. (2) we have the
♦ The plus sign is here used, since we have gone around the figure in a direc-
tion opposite to that followed in the general case
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SURVEYING,
RULE FOR FINDING AREAS FROM RECTANGULAR OFFSETS AT
IRREGULAR INTERVALS.
Multiply t/ie distance along the course of each intermediate
offset from the first by the difference between the two adjaceyit
offsetSy always subtracting the following from the preceding.
Also multiply the distance of the last offset from t lie first by the
sum of the last two offsets. Divide the sum of these products by
two. •
The following is the numerical reduction for finding the
area of the irregular tract shown in Fig. 59.
0£foet
Distance
from A.
Leuffth of
O&t.
Differeoces.
Products.
ch.
ch.
ch.
sq.ch.
B
0.00
1.53
C
1. 21
1.76
- 0.47
- 0.57
D
2.23
2.00
— 0.56
- 1.25
E
3.56
2.32
+ .09
+ .32
F
5.04
1. 91
+ .87
+ 4.38
G
5.75
1-45
+ .91
+ 5.23
H
7.00
1. 00
+ 2.45
+17.15
2 ) 25.26
Area = 12.63 sq. c^s.
= 1.263 acres.
It is evident that an area bounded on all sides by irregular
or curved lines could have a base-line run through it, and off-
sets taken from this line to both boundaries and the area com-
puted by this method. Example 10, p. 235, should be so
computed.
189. The Method by Offsets at Regular Intervals.— If
the intervals between the offsets, or ordinates, are all equal the
computation is much simplified. On the assumption that the
area is a series of trapezoids, we have the
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LAND SURVEYING. 2ig
RULE FOR FINDING THE AREA FROM RECTANGULAR OFFSETS
AT REGULAR INTERVALS.
Add togeiher all the intermediate offsets and one half the end
offsetSy and multiply the sum by the constant interval between
them.
The following rules for finding areas are found from the suc-
cessive orders of differences in each case and may all be derived :
by a rigid development.* They assume that the bounding-line
is curved and that rectangular ordinates have been measured
at uniform intervals from a base-line traversing the figure.
Let the common interval between ordinates be d\ let the
lengths of the ordinates be ^„ A„ A, . . . . A» ; and let the
number of intervals be N,
\. N ^\, -4 = - (A, -f A,), Trapezoidal Rule.
II. iV = 2, A -=• - {h. + 4^, +^«)» Simpson's \ Rule.
III. JV = 3, A = y (A.+ 3*. + ZK + K), Simpson's f Rule.
IV. i\r=4. ^ = ^[7(A. + A0 + 32(>4. + //.)+i2A.].
W.N =6, A = ^^ [A, + A. + A,+ >5.+ 5 (A.+ A,+ A,)-j-A,].
This is called Weddel's Rule. If a quadrant be computed by
this rule, the result is 0.779^" instead of o.^Ssr'y the true value.
When an area, bounded by a base line and two end ordi-
* Sec appendix C.
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220
SURVEYING.
nates, be divided by imaginary lines parallel to the end ordi-
nates and equally spaced, as in Fig. 60, and if the middle ordi-
nates of these partial areas be measured, then il d =z common
width of the partial areas and A,, A,, A„ etc., their middle ordi-
nates, a the first end ordinate and 6 the last one, we have,
approximately,
1. A = dSA,
where 2A signifies the summation of all the A's.
The following rules are, however, more accurate :
II. A = d2A + ~ (^ - A, + * - Ah), Poncelet's Rule ;
or,
[Rule.
III. A = d2A + -{Sa+A,-gA,+S6 + An,i-9A,\ Francke's
72
The various rules above given are often used to determine
areas of irregular figures such as steam diagrams, cross-sections
of structural forms, streams, excavations, etc. The most
ready and accurate means of determining all such areas^ how-
ever, is by means of the planimeter.
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LAND SURVEYING,
221
THE SUBDIVISION OF LAND.
190. The Problems arising in the subdivision of land are
of almost infinite variety. All such problems are solved by the
application of the fundamental principles and relations of
geometry and trigonometry with which the student is supposed
to be familiar. There are, however, two classes of problems of
such frequent application that they will be given in detail.
191. To cut off from a Given Tract of Land a Given
Area by a Right Line, starting from a Given Point in the
Boundary. — In Fig. 55, p. 193, let O be the middle point on
the line AB, from which a line is to be run in such a manner
as to cut off three acres from the western portion of the tract.
We may at once assume that the dividing-line will cut the side
DC in some point X^ whose distance from D is to be found.
First compute the area OAED, using the balanced latitudes
and departures given on p. 199, we have the following:
Course.
Lat
Dep.
•
D. M. D.
Double Areas.
+
-
AG
CD
DE
EA
ch.
- 1.26
(+ 8.07)
- 2.49
- 4.32
ch.
+ 3.30
(+3.10)
- 3.92
— 2.48
3.30
9.70
8.88
2.48
4.16
78.28
22.11
10.71
(- 8.07) (- 3.10)
Sums+ 78.28
— 36.98
2)41.30
36.98
Area = 20.65 sq- chs.
= 2.065 acres.
Here the latitude and departure of the course OD are such
as to make the latitudes and departures balance. The area is
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222 SURVEYIN^G.
found to be 2.065 acres, leaving 0.935 acres to be laid off from
OD by the line OX. It remains now to find the point X.
First compute the length and bearing of the line OD from
Case I., p. 213.
Thus we have
D + rio
Whence 0 = 21® from the table of natural tangents. From
the table of natural sines, we find sin 21° = 0.358.
Hence from eq. (3), p. 213, we have
/sin ^ = D, or 0.358/ = 3.1a
Whence / = 8.66 chains.
The bearing is evidently N. 21° E.
We now have to find the distance DX such that the area
ODX shall be 9.35 sq. chains. Since the area of any triangle
is one half the product of two sides into the sine of the in-
cluded angle (another way of saying it is equal to half the base
into the altitude), we have
9.35 =i(8.66x/>^) sin C>Z?X. . . . . (i)
From the bearings of OD and DX we find the angle ODX
to be 60" 30', hence sin ODX = 0.870, from which we find
DX = 2.48 chains.
The length and bearing of the line OX may be computed
from its latitude and departure, the same as was done for the
line OD above, or we may compute the angle i>C7Jr and length
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LAND SURVEYING, 22 S
OX by solving the triangle DOX. The bearing of OX may
then be found, and the line run from O. There will then be
two checks on the work, viz. : the measured lengths of OX and
DX must be equal to their computed values.
To find the angle DOX, let the three angles of the triangle
be Dy O, and X, and the sides opposite these angles be d^ Os
and Xy respectively. Then we have
tan H^- ^ = J^ tan H^+ <^
This equation gives the angle (X — 0)j whence
(9 = i (jr + O) - i {X- 0\ and X = i (^ + C>) + i (^- O)
Also, d=OX=OD^^^
and o = DX=OD
sin^'
sin O
sin X'
We therefore have the following
RULE FOR CUTTING OFF A GIVEN AREA BY A LINE START
ING FROM A GIVEN POINT IN THE BOUNDARY
Having first surveyed the tract and plotted the same, join
the given point on the plot with the corner which will give the
nearest approximation to the desired area. Compute the
length and bearing of this line, and of the area thus cut off.
Subtract this area from the desired area, and the remainder is
the area to be cut off in the form of a triangle, one side of
which has bearing and distance given, and another side has its
bearing alone given. From these data compute the lengths and
bearings of the other sides, one of which is the line sought.
This line may then be run, and its length measured, as well as
the length of the portion of the opposite boundary cut off, for
a check on the accuracy of the work.
192. To cut oflf from a Given Tract of Land a Given
Area by a Right Line running in a Given Directioa
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224
SURVEYING.
—Let the problem be to cut off 30 acres from the northern
portion of the tract shown in Fig. 57, p. 205, by a line whose
bearing is N. 80° E., or whose azimuth is 260°.*
Pass a line parallel to the required line through the corner
nearest to the probable position of the desired line. Let MBy
Fig. 57, be such a line. Compute the lengths of the lines EM
and MB by Case IV., p. 214.
From the computation, p. 206, we have the following :
Coureeft
Azimuth.
Lengths.
Balanced
I^itudes.
Balanced
Departures.
D. M. D.'s
Double Areas.
BC
CD
DE
EM
MB
205' 39'
112 12
55 00
0 04
260 00
1004 ft.
896
912
+ 9o6fl.
4- 339
— 522
+ 432 ft.
- 834
- 750
2738
2336
752
I
"53
+ 2,480,628
+ 791,804
- 392,544
— 926
+ 234.059
(926)
(1171)
— 926
+ 203
— I
+II53
(+ 723) (-II 52)
2 ) 3,113.021
Therefore to close requires Z = — 723 and /> = + 1152. Area = 1,556,510
sq. fL
= 35.73 ac'i.
From equation (10), p. 215, we have
£M^
D cos 260° — L sin 260**
sin 259° 56'
_ (+ 1 152) (+ .1736) - (- 723) (+ >9848)
"" + .9846
200 -f 712
.9846
= 926 ft.
* In this problem it would have shortened the operation somewhat if the
meridian of the survey had been taken parallel to the dividing- line. The bear-
ings could have all been changed to give angles from this meridian, and original
computation made from these new bearings.
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LAND SURVEYING. 22^
Whence from eq (9), we have
sin ;2uo
-4- f I C5 — (c\if%\ ( — YX)I 0
= 1 171 ft.
"■ + .9848.
Inserting these values of the lengths of the courses EM
and MBy we can compute the area BCDEM. This is found to
^c 3573 acres, or 5.73 acres too much. The problem now is
to pass a line north of MB and parallel to it, so that the area
included between the parallel lines and the intercepted por-
tions of £/^and BC shall be 5.73 acres, or 249,710 sq. ft. Let
OO' be such a line. This line can be run when either MO or
BCy is known. It is best, however, to compute both these
distances, using one for a check. To find these distances.
Let X = perpendicular distance between the parallel lines
MB and 0(7.
Let angle EMB = EOa = 0,
and angle OO'B = 0.
Then we have
Area MOO'B = MB, ;r - i^r* cot ^ + ix" cot 0
=^MB.x + ix' (cot 0 — cot ^. • . (i)
Since ^and 0 are known angles, their cotangents are known
quantities in any case. So, for simplicity, let
(cot 0— cot e) = K\
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226 SURVEYING,
also, let the distance MB = A
and area MOO'B = A.
Then the equation becomes
A^Dx + ^Kx'^ (2)
..2D _ 2A
=--^{±i^2AK+ir^D) (3)
That sign of the radical is to be used which will give a
positive value to x. The other sign would give the value of
X to be used in laying off the given area on the opposite side
of MBy provided the sides OM and O'B were continuous in
that direction.
Using equation (3) for the problem in hand, we have
^=79° 56';
0=54° 21';
A = 249,710 sq. ft.;
Z?= 1171 ft.;
K= 0.7172 - 0.1775 = o 5397;
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LAND SURVEYING. 227
whence x = ^^^ (± V26^,S37~+h37iMi - 1 170
= 203.6 feet.
We can now find MO and B(y from
if(7 = -^^ and ^(7' = -t^;
sin cr sin 0
whence MO = 206.8 feet, and BO' = 250.6 feet.
The length of the line 00' is
00' = -Af5 + X (cot 0 — cot ff).
We may therefore write the following
RULE FOR CUTTING OFF A GIVEN AREA BY A LINE PASSING
IN A GIVEN DIRECTION.
Having first surveyed the tract and plotted the same, pass
a line on the plot in the required direction through the corner
which will give the nearest approximation to the desired ajea.
Compute the lengths of the two unknown courses bounding
this area, and then the area itself. Subtract this from the
given area, and the remainder is the area which is to be cut off
by a line parallel to the first trial line. This auxiliary area will
always be a trapezoid, whose area, the length and bearing of
one of the parallel sides, and the bearings of the remaining
sides are known. The lengths of these sides may then be
computed, one of the end lengths laid oflf, and the dividing
line run. Measure the length of this line and also of the other
end line for checks.
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228 SURVEYING,
PRINCIPLES AND LAWS BEARING ON THE RESURVEY OF
PRIVATE LANDS *
193. The Problem Stated.— In all resurveys of private
lands, whether for running boundaries, computing areas, or for
parting off or dividing land, it is first necessary to examine the
description of the tract as given in the deed of conveyance,
and then to identify such marks, corners, boundaries, and monu-
ments on the ground as have been used in the description.
The identification of these monuments often taxes the expert-
ness and skill of the surveyor to the utmost, and it is here that
the greatest experience and judgment are required. The orig-
inal monuments, if there were any placed, may have been entirely
lost, and they may or may not have been replaced by others.
If they have been, it remains to be determined how reliable
these secondary monuments are. In the absence of monu-
ments specially set other natural or artificial features may have
been used in the description or have by use acquired the force
and authority of monuments. There may also be gross discrep-
ancies between the position of the monuments and the de-
scription or area named in the deed. There may also be a
controversy between the parties in interest as to the real boun-
daries which the surveyor may be wholly unable to decide.
This much, however, is certain, that any location of a corner
or line by course and distance is likely to be very uncertain
and unsatisfactory, especially where the needle-compass was
used in the original survey, and that every effort should be
made to find some trace of the original monuments, if any were
set, or to decide from all the evidence available, both material
and personal, what the real boundaries are.
The surveyor has no judicial authority to fix or establish
* The rules laid down here are mostly derived from State Supreme Court decis*
ions in cases which arose over boundaries established by the compass and chain, and
hence do not apply so well to city and town surveys made with greater exactness-
See also Appendix G and L
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LAND SURVEYING, 229
anything.* He is simply an expert witness called in to assist
by his knowledge and experience, and with the aid of his sur-
veying instruments, to find the true lines and boundaries. His
acts arc at all times subject to review in the courts, and he should
try and subject his decisions to the same rules and precedents
which are likely to govern the court. He may thus save him.
self from much embarrassment and his client from unnecessary
expense.
Where a surveyor makes gross blunders in his work, show^
ing incompetency, he is often held for any damages which may
result. In many cities licensed surveyors are under a heavy
bond and are held liable for any erroneous locations they may
make.
Surveyors' records and plats should always be complete and
definite, otherwise they cannot be admitted as evidence.
The following propositions are based on judicial decisions
which are thought to have all the force and authority of law,
in the absence of special statutes governing the case :
X94. The Interpretation of Descriptions in Deeds and the Identifica«
tion of Boundaries. — General Ruies,^i, If the description is inconsistent,
insafficient, doubtful, or capable of two or more constructions, the purchaser is ta
be given the reasonable benefit of such defects. That is to say, the grantor \x
required to convey the land under the most favorable legitimate construction
which may be put upon the description the grantor has used to describe it.
But if the intention is evident on the face of the instrument, or if the parties by
their acts have shown a mutual agreement or acquiescence in a certain interpreta.
tioo of the description, this meaning will hold and bind the parties.
2. Where any inconsistency in the description arises from a false or impossible
statement, and by rejecting such evident error the remaining description becomes
consent and possible, then such part should be rejected and the deed allowed to
stand.
Also, when parts of the description are certain and others uncertain, if the
inconsistency can be removed by rejecting one or more of the uncertain portions,
this may be done.
* See the valuable paper, by Chief -Justice Cooley, on ** The Judicial Functions of
Surveyors," read before the Michigan Association of Engineers and Surveyors, and
printed in full in Appendix A. See also Appendix G and I.
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230 SURVEYING.
Again, an entire side which has evidently been omitted from the description
may be supplied if the monuments and area may thereby be made to agree with the
description. If possible, however, every call in the description should be satisfied
by the surveyor in locating the property.*
3. All descriptions in deeds must be construed in the light of what was known
to and in the minds of the parties at the time the description was first written, and
with reference to such plats, facts, and monuments as then existed.
4. The law presumes the deed was drawn with an honest intent to convey the
property. The description must, therefore, be construed, if possible, in such a
way as to make it effectual rather than void.
5. If the lines are fixed by monuments, and these can be clearly identified, and
are used in the description, such lines are the true boundaries, and are to be deter-
mined by a resurvey, even though they differ from the plat, or from the description
given in the deed. Any conclusive evidence of the original location of these lines
and monuments will, therefore, overrule all surveys or other forms of evidence of
where they should have been. If the boundaries were not marked at the time the
plat was made, then the description is to govern, subject to the rules on excess
and deficiency given below.
6. Where land is simultaneously subdivided into numerous tracts, as in the case
of the United States land surveys and in the case of town plats, all the marks,
lines, and monuments set in the original survey for subdivision serve as marks,
lines, and monuments for every tract or lot in the original survey and are some
evidence of the location of each tract or lot. In the absence of monuments mark-
ing the location of a p>articular tract or lot other monuments of the same original
survey may be used, but monuments placed in preceding or subsequent surveys, or in
surveys of adjoining territory not a part of the given subdivision, cannot be so used.
7. In the absence of monuments which can be identified, conclusive evidence
of the original position of such monuments, or of the lines themselves, may set
aside the courses and distances called for in the deed. In short, boundaries may
be proved on such testimony and evidence as may be adduced to establish any other
fact. The surveyor should, however, gfive great weight to the courses and dbtances
called for, as a part of such material evidence.
8. Where streets have been opened and used for a long period and the lines
marked by fences or other material boundary, and these lines have been acquiesced in
without protest, such marks obtain the force and authority of monuments and should
not be disturbed because of any disagreement with the original plat and description.
9. All monuments established in United States land surveys are presumed to
be equally well placed and have equal authority or weight in determining boun-
daries. Thus a quarter-section comer has the same weight as a township or section
corner, even in fixing a township line. Also, section corners set on lines closing
on the north and west sides of townships, though not lying on the original town-
ship lines, should govern the location.
* See also Art. 304, Chap. XII., on City Surveying.
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LAND SURVEYING. 23 1
10. In the United States land surveys the several sized tracts are section, quarter-
section, half-quarter section, and quarter-quarter section. These are presumed to con-
tain 640, 160, 80, and 40 acres respectively, and the Government sold them as contain-
ing these amounts.* The manner of subdividing a section is defined by law (see p.
183), and hence any actual subdivision of section into quarters and a quarter-section
into halves and quarters ag^n is always subject to revision and correction until the
law is satisfied, except that the quarter-section comers, planted on the section lines
by the United States Deputy Surveyors, cannot be changed. Other subdivisions
than those here named are not subject to any law as to the methods to be pursued.
11. In order that monuments may control when inconsistent with the courses
and distances used in the description, they must be mentioned in the deed. If not
so mentioned, or if mentioned but not capable of identification, then the courses
and distances govern.
Particular Cases. — i. Where land is described as being "owned and occu-
pied,' the actual line of occupation is a material call of the deed.
2. Where a boundary line is defined by distance and terminus at a known point
or line, this known terminus fixes the length of the line. If the position of the
terminus is uncertain the distance governs.
3. Acquiescence in a given boundary erroneously placed does not alone fix the
boundary if the issue has not arisen, and jf the fact of such error has not come to
the attention of the parties.
4. A course described as running from point A to point B is presumably a- straight
line but if not so stated it may be construed as a crooked or curved line if it is
understood to follow some natural feature of the landscape.
5. The terms "southerly," "westerly," etc., are to be construed as meaning
due south, due west, etc.. if there is nothing to indicate the contrary. Also where
terms of approximation are used, such as " about," "%s near as may be," and the
like, if the exact figures given fit the case and satisfy the description as well as any
other, the interpretation is limited to the figures stated.
6. Where the described boundaries are complete and consistent, but inconsis-
tent with the stated area, the boundaries hold as against the area. If the bounda-
ries are doubtful the area may control.
7. Where the call is simply for a given area without dimensions it must be
taken in the form of a square if such a rendering is not excluded by some other
condition. If one side of the tract is given in line and distance it must be laid out
as a rectangle upon such side.
8. In case of an accompanying plat showing monuments, courses, and distances,
which plat is referred to in the deed, but the description not repeated in the text,
the description will hold.
* Whenever these legal subdivisions are mentioned as such in deeds of convey-
ance, as "the S. E. 40 acres of the N. E. quarter of section 10," etc., nothing
more is intended than simply " the S. E. quarter of the N. E. quarter of section
10." tic, and the conveyor cannot be held for the full area named^
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232 SURVEYING.
9. Land bordering on a public highway usually takes to the center of the
highway, unless expressly stated to the contrary. This do3s not apply to city and
town lots where the streets have been reserved in the original plat.
10. All surveys and descriptions should close when platted, and the surveyor is
usually at liberty to use his judgment in correcting either course, or distance, or
both, where no monuments are identified with sufficient certainty to give them
authority. Where monuments remain they control the boundaries so far as they
go, in which case the description is not obliged to close, or if it is made to close
the points marked by monuments must not be disturbed.
In closing a survey or description, there being no other guide, that method will
be used which will convey ihe greater quantity of land in accordance with the prin-
ciple that the description is to be construed in favor of ihe purchaser.
11. Where the bearings of all the courses are given and one course can be
identified with certainty, the declination (or *' variation'') of the needle used in
the original survey should be found by setting up on this course and the declina*
tion thus found used for all the courses.
12. When a course is defined as starting at a given point on a navigable stream
or travelled highway and running a certain distance to another point on said stream
or highway, the distance is to be measured along the line of the bank of the stream,
or along the highway, and not in a straight lice, unless it is specifically so stated.
If the stream is not navigable and the presumption is against it being a boundary,
the distance is to lie measured in a straight line.
But where a tract of iand is described as bordering on or fronting a certain dis-
tance on a stream, in the absence of other controlling facts such distance must be
measured in a straight line between the extremities of the opposite boundaries.
195. Water Boundaries and Meandered Lines. — i. Meandered lines
on the United States land ^rveys were run for the purpose of outlining lakes and
rivers, and are in no sense boundary lines. They served for computing the areas of
the fractional quarter-sections which were used in the first sales by the Government,
but the real boundary is the center of the stream if not navigable, and the line of
ordinary high water, or line of vegetation, if navigable. In the case of lakes and
ponds described as boundary lines the ownership is to the water's edge.
2. In extending side boundaries beyond the meandered lines to the river-bank
or lake-shore, such extensions beyond the meandered lines should run at right angUs
to the short. This rule also applies to city lots and to all lands fronting on bodies
of water. An exception to the rule is the following :
3. When the waters of a lake recede from drainage or any other cause, or when
a river or creek shifts its course, the accretion or "made land " belongs to the
abutting property and should be divided in proportion tu the original lengths of
water-frontage. If the land thus acquired is the valuable consideration then the
original side boundaries are to be extended so as to divide the new area among
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LAr^D SURVEYING. 233
the abutting tracts in proportion to their original frontages ; but if the length of the
new frontage is the desirable thing, then the new line of frontage is to be saved to
the original tracts possessing such privileges, in due proportion. In either case the
extension of the side boundaries will usually involve angles at the meandered or
original water-line and these extensions will, in general, run nearly at right angles
to the new shore line.
4. A ** bank*' of a stream is the continuous line where vegetation ceases. A
' * shore " is the exposed ground below the bank line.
6. The rights of ownership extend to the centre lines of non-navigable streams
and lakes, but only riparian rights obtain in the beds of navigable streams or
hikes.
6. Where land is specifically bounded by ** the bank/' or ** along the bank " of
a stream or lake, these words will exclude all ownership of the bed of such stream
or lake. Whether they would exclude riparian rights also would depend on the
circumstances of the case and on the understanding of the parties.
In computing the area of a survey the terms ** from," " to," or "with" the
bank of a stream mean to low water mark.
7. In the case of meandered river-banks or lake-fronts on the United States
land surveys, the computed areas included only up to the meandered line, all out-
side of that belong to the tract by a natural right. Hence in any subsequent sales
of the tract the area should only be computed to the meandered line unless the con
veyance specifically calls for an extension ** to" or ** along" the shore or bank, in
which case the area would be computed to low water mark, as above stated.
8. Similarly, when an area is to be laid off from a tract bounded in part by a
meandered line, this area should be computed only up to the meandered line unless
otherwise specifically stated.
9. Islands in streams unsurveyed by the United States and unappropriated
belong to the abutting land on that side of ihtjilum aqua or the central thread of
the low water channel on which the island itself lies.
196. Surplus and Deficiency.*—!. Surplus or deficiency, either of dis-.
tances or of areas, does not invalidate a conveyance.
2. In the case of contiguous tracts where no monuments were established, or
where they have been lost, the purchasers receive their full measure of ground, in
the order of purchase from the original owner, the last purchaser receiving the
surplus or losing the deficiency.
In the case of city lots, sold by number, any surplus or deficiency found on the
ground should be divided proportionally among all the lots affected, but a suit for
a proportionate pait of the surplus would probably not hold, and in case of defi-
dency, if all but the last purchaser should take his full portion, the last man would
probably have to content himself with the remainder, and pay for only so much as
he gets.
*See also Art. 305, Chap. XII., on City Surveying.
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234
SURVEYING.
EXAMPLES.
X. Compute the area, plot the survey, and determine error of closure from the
following field-notes :
Sution.
Bearing.
Distance.
A
S. 46i" E.
20.00 ch.
B
S. 74i E.
30.95
C
N. 33i E.
18.80
D
N. 56 W.
27.60
E
W.
21.25
F
S. 5if W.
13.80
Answer
{Area = 104.4 ± acres.
Error of closure = i in 201.
This being a compass-survey, the errors in latitude and departure must be
distributed in proportion to the lengths of the courses, regardless of their bear-
ings, or according to Rule i, p. 200. If the errors in the bearings (or deflection
angles) had been very small as compared with the errors in measuring the dis-
tances, as is the case when the deflection angles are measured with a transit,
then Rule 3, p. 201, should have been used.
2. Find the area and error of closure from the follovnng field-notes :
Sution.
Bearing.
Disunce.
A
E.
130 rods.
B
N. 8°E.
137
C
N. 81 W.
186
D
S.
54
E
S. 36 w.
125
F
S. 45 E.
89
G
N. 40 E.
70
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LAND SUR VE YWG.
235
What would be the resulting difference in area from the use of Rules i and 3 ?
3. In Example i, suppose the length and bearing of the first course were
unknown. Let these be found as in Case I., Art. 186.
4. Suppose the length of course A and bearing of B are unknown in same
example. Compute by Case II.
5. Let the first two bearings be unknown. Compute them by Case III.
6. Let the lengths of the first two courses be unknown. Find them by Case
IV.
7. Let it be required to cut off twenty-five acres from the west end of the
tract given in Example i by a line passing through a point on the course BC 2X2^
distance of ten chains from B. Find the length and bearing of the division-line
and the other intersecting point on the boundary.
8. Let it be required to divide the tract given in Example i into three equal
portions by north and south lines. Find the length and points of intersection of
such lines with the boundary-lines.
9. Compute the coordinates of the comers of the tract given in Example i,
taken with reference to a point 35 chains directly south of //, and then compute
the area of the tract from these coordinates by the formula given in Example i.
This area should, of course, be the same as that obtained by any other method
where the same balanced latitudes and departures are used.
10. An irregular tract of land has a straight line run through it and rectangular
offsets taken to the boundary. Find the area of the tract from the following
notes :
Distance.
Width.
ch.
ch.
0
2.35
10
8.42
M
12.60
20
11.38
25
10.75
28
6. ,5
30.50
0.00
Is it significant whether or not this tract lies on both sides or wholly on one
si ie or the base-line ?
II. Compute the area of the tract of which the following are the field-notes.
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236
SURVEYING,
Tht ectangular offsets are taken on both sides of a straight axial line, R signify,
ing right and L left.
Distances.
Side.
Width or
Disuuices.
Side.
Width or
ch.
ch.
ch.
ch.
0
R
4.23
18
R
15.80
0
L
0.00
20
L
5.00
5
R
7.16
25
R
12.20
750
L
3.45
30
L
2.62
10
R
12.68
30
R
6.48
10
L
6.00
30
L
0.00
12
R
10.75
1
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CHAPTER VIII.
TOPOGRAPHICAL SURVEYING BY THE TRANSIT AND
STADIA.*
197. A Topographical Survey is such a one as gives not
only the geographical positions of points and objects on the
surface of the ground, but also furnishes the data from which
the character of the surface may be delineated with respect to
the relative elevations or depressions.!
198. There are three general methods of making such a
survey.
First, with a compass (or transit) and chain,. to determine
geographical position, and with a level for obtaining relative
elevations.
Second, with a plane-table, either with or without stadia-
rods.
Third, with a transit instrument and stadia rods.
The first method is very laborious, slow, and expensive. It
is therefore not adapted to large areas. The second method
has been more extensively used for this purpose than any
other. The use of the plane-table is fully described in Chap-
ter V. This method is giving place, however, to the third,
which has been in use in America since about 1864, when it
was officially adopted on the United States Lake Survey.
The system was first used in Italy about 1820. In what fol-
lows, the third method will alone be described.
*The word "stadia" is Italian and was originally used to designate the
rod used by the invenlnr of the m«»thod. It is now too firmly estabh'shed to
be changed. On the U. S. Coast and Geodetic Survey the word "telemeter"
is used in place of "stadia," but this, which very properly means distance-meas-
urer^ has been appropriated for other appliances used for measuring at a dis-
tance, as temperature, for example. It would therefore seem that "stadia**
is the better word to use.
t See also Appendix G. r^ \
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238 SURVEYING.
199. The Principle of the location of points by the transit
and stadia, both horizontally and vertically, is that of polar
coordinates. That is, the location of the point geographically
is by obtaining its angular direction from the meridian through
the instrument, which is read on the limb of the transit, and
its distance from the instrument, which is read through the
telescope on the stadia-rod which is held at the point. This
distance is found by observing what portion of the image of
the graduated rod is included between certain cross-hairs in the
telescope. The farther the rod is from the instrument, the
greater is the portion of the rod's image which falls between
the cross-wires.
For elevation, the vertical angle is read on the vertical circle
of the transit, when the telescope is directed towards a point
of the stadia-rod as far from the ground as the telescope is
above the stake over which it is set. The tangent of this
angle of elevation, or depression, into the given horizontal dis-
tance is the amount by which the point is above or below the
instrument station.
In this way, both the chain and levelling-instrument are dis-
pensed with, and the slow and laborious processes of chaining
over bad ground, and levelling up and down hill, are avoided.
The horizontal distances are obtained as well, in general, as
by the chain; and the levelling may be done within a few
tenths of a foot to the mile which is amply sufficient for topo-
graphical purposes.
THEORY OF STADIA MEASUREMENTS.*
200. Fundamental Relations. — In Fig. 61 let LS be any
lens, or combination of lenses, used for the object-glass of a
telescope.
• For a good description of A New Prismatic Stadia see Jour, Asso, Eng, Socs,
vol. xiii.. p. 43 (Jan., 1894). One half of the telescope objective is covered with
a prism which causes two portions of the rod» rays from which form a fixed angle
at the objective, to coincide in the image, as in a sextant, thus dispensing with the
use of stadia wires.
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TOPOGRAPHICAL SURVEYING.
239
Let ^2^2 be a portion of the object (in this case the stadia-
rod), and let A^B^ be its image. The point of the object A^ has
its image formed at ^i, and so with B^ and B^.
Let F be the position of the image for parallel rays, or for
an object an infinite distance away; and let C be the centre of
the instrument, or the intersection of the plumb-line, extended,
with the axis of the telescope.
Let E^ and E^ be the *' principal points,"* and let the
distance FE^ = f (focal length),
Q^ Z. f I (conjugate foci),
Afi^ = i (for image, intercepted portion),
Afi^ = s (for stadia, intercepted portion).
Then, since A^E^ is parallel to A^^, and B^E^ is parallel to
B^E^ we have *
A,B, : A^, :: IE, : OE^,
or, i:s ::/,:/,. .
Also, from the law of lenses we have
(I)
* As optics is generally taught in the English text-books, Ei and Et are
made to coincide in a point at or near the centre of the lens; and this is called
the "optical centre." The "principal points'* of the ordinary objective fall
inside the surfaces of the lens, but they never coincide. The ordinary theory
is sufficiently approximate for the development of stadia formulae but it saves
confusion to make the conditions rigid, and it is equally simple.
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240 SURVEYING.
f^jry (^>
On these two equations rests the whole theory of stadia
measurements.
Since the distance FE^ =/= focal distance, is a constant
for any lens br fixed combination of lenses, we see from equa-
tion (2) that if the object P approaches the lens the distance
/, is diminished, and therefore f^ must be increased ; that is,
the image recedes farther from the lens as the object ap-
proaches it, and vice versa.
If the extreme wires in the reticule of the telescope be sup-
posed to be placed at A^ and B^ in the figure, then A^E^B^ is
the visual angle which is equal to AJE^B^, But as the image
changes its distance from the objective as the object is nearer
to or farther from the instrument, so the reticule is moved
back and forth,* for it must always be in the plane of the
image. Therefore lE^ =/, is a variable quantity, while A^B^
is constant for fixed wires. Therefore the visual angles at E^
and E^ are variable.
If these angles were constant, the space intercepted on the
rod, and the distance of the rod from the objective, would be in
constant ratio. Since this is not true, we must find the rela-
tion that does exist between the distance Efi and the space
intercepted on the rod, A^B^.
From equation (i) we have
i — i
fnf:
I
but from equation (2) -r = -7 — rr
* If the objective is moved in focusing it does not appreciably affect these
relations.
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TOPOGRAPHICAL SURVEYING. 24.
Equating these two values of — , we have
^ _ I I
or
A-=jS + f; (3)
that is, the distance of tite rod from the objective is equal to
the intercepted space in the rod multiplied by the constant
/
ratio -., plus the constant /, where /is the focal length of the
t
objective, and i is the distance between extreme wires. If
the distance between the extreme wires be made o.oi of the
focal length of the objective, then the distance of the stadia-
rod yr<7i« the objective (rigidly from E^ is a hundred times the
intercepted space on the rod, plus the focal length of the ob-
jective.
Again, if a base be measured in front of the instrument,
with its initial point a distance f in front of the object-glass of
t/ie telescope, then the rod may be held at any point on this
base-line, and its distance from the initial point, and the space
intercepted by the extreme wires, will be in constant ratio.
The lines AJ^' and BJ^' in Fig. 61 show this relation, for
they are the lines defining the space on the rod which is inter-
cepted by the extreme wires as the rod moves back and forth.
Evidently the rod cannot approach so near as F , for then the
image would be at an infinite distance behind the lens. Usu-
ally the extreme position of reticule does not correspond to
a position of rod nearer than ten to fifteen feet.
It must be remembered that any motion of the eye-piece,
with reference to the image and wires, is only made to accom-
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242 SURVEYING,
modate different eyes, and has no effect in changing the rela-
tion of wire interval and image. The eye-piece is simply a
magnifier with which to view the image and wires, but in all
erecting- instruments it also reinverts the image so as to make
it appear upright. The effect of the eye-piece has no place in
the discussion of stadia formula;.
If the distance of the stadia is to be reckoned from the
centre of the instrument, which it usually is, and if this dis-
tance = d, and the distance from the centre of the instrument
to the objective {CE^ in Fig. 6i) = r, then we have, from (3),
d--f.^-c = is^f^c. (4)
Since/, 1, and c are constant for any instrument, we may
measure / and c directly, and then find the value of i by a
single observation. Proceed as follows:
1st. Measure the distance from the centre of the instru-
ment (intersection of plumb-line with telescope) to the objec-
tive, and call this c,
2d. Focus the instrument on a distant point, preferably the
moon or a star, and measure the distance from the plane of
the cross-wire to the objective, and call this/!
3d. Set up the instrument, and measure the distance /+ c
forward from the plumb-line, and set a mark. From this mark
as an initial point, measure off any convenient base, as 400 feet.
4th. Hold the rod at the end of this base, and measure the
space intercepted by the extreme wires. If we call the length
of this base b, and the distance intercepted 5, then we have,
from equation (3),
or * = l/ (5)
b^
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TOPOGRAPHICAL SURVEYING. 243
Here we have the value of i in terms of known quantities.
If it is desirable to set the wires at such a distance apart
s
that -T will be a given ratio, as y^, then i must equal o.oiy. It
is possible to set the wires by this means to any scale, so that
a rod of given length may read any desired maximum distance.
If it is desired that — should be determined with great ac-
curacy for a given instrument, with wires already set, so as to
have a coefficient of reduction for distance, for readings on a
"od graduated to feet and tenths, for instance, proceed as fol-
ows:
Make two sets of observations for distance and intercepted
nterval. The distances should diflfer widely, as 50 feet and
f 00 feet, or 100 feet and looo feet, according to the length of
rod used. The shorter distance should not be less than 50 feet,
and the longer one not more than 1000 feet with the most
favorable conditions of the atmosphere. The distances are to
be measured from the centre of the instrument. Make several
careful determinations of the wire interval at each position of
the rod, and take the mean of all the results at each distance,
and call that the wire interval, s, for that distance, d. We then
have two equations and two unknown quantities, these latter
/
being - and (/+ c) in the formula^ equation (4),
Here the d and $ are observed, and ~ and (/+ c) are found.
Knowing these, a table could be prepared giving values of d
for any tabular value of s for that instrument.
This applies to the reading of distances from levelling-rodr
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244 SURVEYING.
Some engineers prefer, in this case, to observe the wire
interval for various measured distances, from the shortest to
the longest, to be read in practice, and prepare a table by inter-
polation. If the observed positions are sufficiently numerous,
this method should give identical results with those obtained
by the use of the formula. The two methods may be used to
check each other.
From equation (4) we see that the distance of the rod from
the centre of the instrument is a constant ratio (-) times the
intercepted space on the rod, plus a constant {/-{-c).
If diagrams or designs be drawn on the stadia-rod to the
scale -^j or so that 10 X ^ yards on the rod would correspond
to 10 yards in distance, and if the rod were decorated with
symbols of this size, then the distance of the rod from the
instrument could be read at once by noting how many symbols
were intercepted between the wires. To this distance must
then be added the small distance (/+ ^)» which is from 10 to
16 inches in ordinary field-transits. On all side-readings, taken
only to locate points on a map, this correction need not be
added, as one foot is far within the possibilites of plotting.
200a. The Use of an Interval Factor. — The practice of
graduating a rod to fit the stadia wires of a particular transit
has been found less accurate than the system which employs
an interval factor with the rod graduated into standard units.
The former method is objectionable, also, for the following
reasons: first, the cost of re-graduating and re-painting the rod
when the interval of the transit changes, as when new wires are
inserted, or from other causes ; second, such rods cannot be
interchanged among other transits, or old rods used with new
transits ; third, such non-standard rods could not be used in
leveling without computing laborious corrections.
All these objections are overcome by the use of rods grad-
uated into standard units and the use of an interval factor
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TOPOGRAPHICAL SURVEYING. 245
( AT = - ) to multiply all stadia readings on the standard rod
to obtain the true stadia distance, viz., what the rod reading
would have been if K had been equal to 100.
This method has failed of adoption because of the seem-
ingly large amount of extra computation involved. By the
use of a reduction table, however, the labor involved is very
slight, while the advantages are many. Such a table would I
be prepared as follows : From the adopted interval compute
the distance corresponding to each stadia reading given in the
columns marked Stadia in the following reduction table, and
having added the f -\- c to each of such computed distances,
place the sums in second column opposite the proper stadia
reading ; / + ^ should not be added to readings of less
than o,i.* Thus with an interval factor if= 104.80, and
/+ c = 0^.32 the distance corresponding to a stadia reading
of 0.1 would be 10.48 + 0.32= 10.80. Remembering that
(leaving out they+ c) the distances are proportionate to the
stadia reading, the work of computing the table can easily be
done in an hour's time.
It is assumed in this explanation that the interval was
determined by measuring a base line whose zero was /*-[" ^ in
front of the centre line of transit as explained on page 242.
If the zero of the base line coincided with the centre line of the
transit (plumb line), the construction of the table would be
just the same, except that there would be no/+ c additions
to the column headed Distance.
How to Use the Table. — If, for example, the notes contained
a stadia reading of i"'.96o, to find from the table the true dis-
* This is assuming chat stadia readings of less than o.i will never be
taken, except as a. pari of a much longer reading, as, for example, .06 forms
a part of 1.96 ; so that if /+ ^ were added in the table both to the dis-
tances corresponding to 1.9 and .06 of the rod reading, the resulting sum
would have (/+ c) added twice instead of once.
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245^
SURVEYING.
tance, viz., what the stadia would have read if the interval
factor had been loo.o, you enter the table with 1.9 as an
argument and take out the distance 199.44 corresponding to
it ; also opposite .06 find 6.29, which add to 199.44, giving
205.73 metres as the true distance (and including in it the
STADIA REDUCTION TABLE.
/+^ = 0^.32 A' =104.80
stadia
Reading:.
DisUnce.
Stadia
Reading.
Distance.
Sudia
Reading.
Distance.
.001
.10
.05
5.24
.9
94.64
.002
.21
.06
6.29
I.O
105.12
.003
.31
.07
7.34
I.I
115.60
.004
.42
.08
8.38
1.2
126.08
.005
.52
.09
9-43
1.3
136.56
.006
.63
.10
10.80
1.4
147.04
.007
.73
.2
21.28
1.5
J57.52
.008
.84
•3
31.76
1.6
168.00
.009
.94
.4
42.24
1.7
178.48
.01
1.05
.5
52.72
1.8
188.96
.02
2.10
.6
63.20
1.9
199.44
.03
3.14
.7
73.68
2.0
209.92
.04
4.19
.8
84.16
etc.
etc.
With a little practice it will be found that the field-notes
of an entire day may be reduced in this manner in fifteen
minutes or less.
If the interval factor should happen to be nearly equal to
100, it would not be necessary to reduce the side shots, as on
the usual scales of maps small differences would not be ap-
preciable on the map ; but even in such a case it w^ould be well
to reduce the stadia shots of the main traverse line, because
the omission of such corrections would introduce an accumula-
tive error which might vitiate the accuracy of the entire map.
201. A Simple and Accurate Way to Determine the
Wire Interval of a Transit. — In all topographic surveys
extending over a considerable area a triangulation control is
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TOPOGRAPHICAL SURVEYING. 24$*
always found necessary. The accurately known distances
computed from the triangulation can be made to give the very
best determination of the stadia interval without any further
field-work for this purpose.
In taking topography by the transit and stadia method, it
is usual to begin and end the network of traverse lines (upon
which the topography is made to depend) at triangulation
stations.
If the interval of the transit used be assumed to be loo,
then the distance between the two adjoining triangulation
stations, forming the terminals of the traverse, can be com-
puted from the distances and bearings of the traverse. Then
ICO times the ratio of the true length, as determined by
triangulation, to the distance so computed is the stadia inter-
val of the transit ; e.g., if the distance between two triangula-
tion stations, as computed from the stadia traverse, should be
15,488 feet (assuming the stadia interval to be 100), and the
true triangulation distance between the same points was
15,698 feet, then the assumed interval of the transit should
be multiplied by the ratio of 15,698 : 15,488 = 1.0135, giving
a true interval of 1.0135 X 100 (the assumed interval) =
10K35.
Besides being extremely simple, this method has the added
advantage of having been determined by the person who did
the instrumental field-work of the survey,* and under the same
conditions regarding time of day, weather, etc., as governed
the topographic field-work, thereby insuring a result free from
any systematic errors. The resulting interval would also be
free from accidental errors unless some blunder was made in
the rod readings during the field-work. To guard against such
a possibility it would be well to make two or more computa-
• In order to eliminate a possible error from personal equation, which
has sometimes been found to exist. 0
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245^ SURVEYING.
tions of the interval, by the above method, using different lines
in the triangulation. The agreement of the several deter-
minations so made would insure freedom of error in the result-
ing interval.
Note. — In the above method it is assumed that either
there was no azimuth error in the traverse line, or that it had
been duly corrected or distributed before the distance between
triangulation stations had been computed.
202. How to Prevent Systematic Errors in Stadia
Measurements. — When the wire interval of a transit is accu-
rately determined, stadia measurements are subject only to the
accidental errors of reading the rod. According to a well-
known law only the square root of such errors remains
uncompensated.
As a matter of fact, the results of stadia surveys show
much larger errors in measurement, in some constant direc-
tion, pointing conclusively to an incorrect interval determina-
tion or rod graduation. Professor L. S. Smith has shown *
that the failure to secure a correct wire interval has been due
to a lack of care in securing as near as possible the same con-
ditions for the interval determination as are met in the field
measurements. This requirement is important because, as he
has experimentally proved, the effect of refraction is much
greater near the ground in the strata of air traversed by the
lower line of sight than it is in the strata a few feet above
traversed by the upper line of sight (see Fig. 6i^). This causes
the actual rod reading to differ from what it would have been
had the air been homogeneous. This difference in the amount
of refraction changes in amount at different hours of the day,
giving a slightly different rod intercept for the same distance.
* Sec Bulletin of the University of Wisconsin, Engineering Series,
Vol. I, No. 5, 1895; also Engineering News, Vol. XXXIII, p. 364.
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TOPOGRAPHICAL SURVEYING,
24Sd
In the typical curve shown in Fig. 6ia it will be seen that the
rod intercept is least during the middle of the day, and
greatest in morning and evening.
If an interval was determined by rod readings taken on
the base line near noon, smaller rod readings would result than
* W
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It 1* 1 3
It * X< 4
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. It
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Fig. 6z«.
Fig. txb.
the average readings for the day. As a result of the rod read-
ings, $y being too small (since the interval K equals — ), K
would be too large, and therefore measurements made with
such a value of the interval (or with a rod divided under these
conditions) would as a rule be excessive.*
Just the opposite eflfect would result with an interval made
in the early or late part of the day, viz., the value of K ob-
tained would be too small. However, if several determinations
or tests be made, distributed over several hours of the field
day, and better still on several days, the average of them all
would give an interval comparatively free from systematic
error.
* For a good example of such a case read the report of the St. Louis
Topographic Survey, Journal of the Association of Engineering Societies,
Vol. XII, p. 20. The average error of every sight on this survey was
about + 1.5 f«et.
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246 SURVEYING,
As a result of these experiments the following rules for
determining the wire interval or for graduating the rod become
imperative for the most accurate work. These rules apply to
any or all the methods previously described.
1. Every instrument man slwuld determine for himself his
wire interval {or make the observations for graduating his rods).
2. Determine the wire interval for various distances (but only
between the limits^ expected in the field-work), and for several
hours distributed through one or more days, on a base line which
does not differ radically from the country to be surveyed.
3. For a radical change of field or season conditions, redeter-
mine the wire interval or rod graduation.
4. Avoid reading the lower cross-wire near the ground, either
in the interval determination or in the field-work, but the interval-
determination readings should agree in this respect with the
average field practice.
It is the evident purpose of these rules to insure as far as
possible that every condition obtaining during the test shall
be as similar as possible to the conditions expected or planned
for the field-work. The experiments described in this article
have unquestionably proved that if these rules be followed the
accuracy obtainable will be very considerably increased and
the stadia method thereby made even more valuable than it
has been in the past.
' 203. Adaptation of Formulae to Inclined Sights. — The
discussion given in Art. 2CO is applicable to horizontal sights
only.
If the rod be held on the top of a hill, and the telescope
pointed towards it, the reading on the rod will give the linear
distance from instrument to rod, provided the rod be held per^
pendicular to the line of sight. As it would be inconvenient to do
this, let the rod be held vertical in all cases. When the line of
sight is inclined to the rod, the space intercepted is increased
in the ratio of i to the cos of the angle with the horizon.
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TOPOGRAPHICAL SURVEYING.
247
Thus, the space A'B' (Fig- 62) for the rod perpendicular to
the line of sight becomes AB for the rod vertical But
A'B' = ABcos V approximately.
Fig. 62.
Let A'B' = r', the reading on the stadia for perpendicular
position;* and
Let AB = r, the actual reading obtained for a vertical
position.
Then r^ = r cos v.
f ^
But in equation (4) we have"^ s=± H^ and therefore r' -j- ^
+ /is the distance CO' \ whereas the distance on the horizon-
tal, CO, is generally desired, and for this we have
CO = d— CO' cosz/ = (/ + ^ +/) cos V
= r cos^ V'\'{c +/) cos V. (7)
This is the equation for reducing all readings on the stadia
to the corresponding horizontal distances.
The vertical distance of O' above O is equal to CC sin V.
* By '* reading on the stadia ** is meant the distance CO^ as read off from the
rod by means of its diagrammatic graduation, as described on page 255.
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?48 SURVEYING,
But Ca = ^^4-/+ c = r cos v+Z+c,
nence
(?d?' = A = r cos vsin v + (/+ c) sin v
= ir sin 2v + (/+ f) sin v, (8)
Equation (8) is used for finding the elevation of the point
on which the stadia is held above or below the instrument sta-
tion.
204. Table V.*gives the values d and h computed from
these formulae for a stadia reading of 100 feet (or metres, or
yards), with varying angles up to 30°.
It will be noted that the second term fn the right member
of equations (7) and (8) is always small, and its value depends
on the instrument used. The values of this term are taken
out separately in the table ; and three sets of values are given
of(^+/), — viz., 0.75 feet, i.oo feet, and 1.25 feet. If the
work does not require great accuracy, these small corrections
may be omitted.
The use of the table directly involves a multiplication fot
every result obtained. Thus, if the stadia reads 460 feet, the
angle of inclination 6° 20', and we have/+r = i foot, then
d = 4.60 X 98.78 + 0.99 = 4554 feet,
h = 4.60 X 10.96 + o.ii = 50.53 feet.
and
The table is not generally used for reductions for rf when the
angle of elevation is less than 3 to 5 degrees. When z/ = 5**
44', this reduction amounts to just one per cent. When an
error of I in 100 can be allowed, then the reduction to the
* See also the Colby Slide-rule, p. 265.
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TOPOGRAPHICAL SURVEYING, 249
horizontal would not be used under 6°. If the second term in
^ _[-y be also neglected, these two errors tend to compensate ;
and if ^ +/for the instrument used is I foot, and both these
corrections be omitted, they do exactly compensate when the
stadia reading is 100 feet, vertical angle 5^ 44'.
<«
(«
200
«
«
4»Q4'.
<«
«
300
««
«<
3° 20'.
<•
M
400
«<
«
2° 52'.
M
«
SOD
««
«c
2° 32'.
M
<«
1000
««
«(
1° 46'.
«
«<
2000
«c
«(
1° 18'.
Therefore the reduction to the horizontal need never be
made when v is less than 2°, and it generally may be neglected
when V is less than 6®.
In obtaining the difference of elevation, A, the term in
c +yniay be omitted for all angles under 6° if errors of o.i
foot are not important. For elevations on the main line, how-
ever, this term should always be included.
In practice, therefore, the tables are mostly used to obtain
the difference of elevation from the given stadia reading and
angle of elevation.
PORRO'S TELESCOPE.
205. The Reading Angle with Vertex at the Centei* of
the Instrument — In 1823 Mr. Porro, a Piedmontese officer,
and afterwards a professor at Milan, invented a telescope
which brings the vertex of the reading angle, A^F^B^^ Fig. 61,
to the center of the instrument, and so gives the true reading
for all distances, without the {c-^-f) correction, which must
always be applied with the ordinary telescopes. Although
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250
SURVEYING.
this is not a very important matter
in stadia work, yet because of this
slight correction, or source of error
when not applied, many engineers and
surveyors have heretofore declined to
use the stadia methods at all. The
great advantages of these methods are
coming to be better known, however,
and soon the demand for Porro's tele-
scope may warrant its manufacture for
ordinary transits. It is not now (1890)
made anywhere in America. This
telescope would serve as well for all
other purposes, and although really
little better for stadia work, it removes
an objection which has, more than any-
thing else, caused the stadia methods
to be generally ignored.
The construction of the telescope is
shown in the accompanying figure.
The lens at O is the objective,
having a longer focal length than
the ordinary objective. At P is an
auxiliary len^ by means of which all pen-
cils of rays originating on the reading-
angle lines, CA^ or C5„ are brought to
a focus somewhere on the parallel lines
mA^ and nB^y respectively. In the figure
only one such pencil of rays is shown,
which emanates from B^. The cross-
wires are at A^ and 5„ and since all
pencils of rays originating on the read-
ing-angle lines, which now meet at the
^ center of the instrument, will be brought
to a focus on horizontal lines through
the cross-wires, it follows that the inter-
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TOPOGRAPHICAL SURVEYING, 2$ I
cepted space on the rod is always proportional to the distance
of the rod from the center of the instrument.
The point Fp is the principal focus of the lens /*, its focal
length being much smaller than that of the lens (9, since its
principal focus is at a considerable distance back of P, as at
F^. The point F^ is the position of the principal focus of O in
front, and is the point where the reading angle has its vertex
with the ordinary telescope, as shown in Fig. 6i. The points C
and Fp are conjugate foci of the lens O, An image will always
be formed to the right of P, even for an object nearer to the
objective than F^, The movements of the objective (or eye-
piece) for focusing at different distances is less for this tele-
scope than for the ordinary telescope. The relative position
of the lenses O and P is fixed, and they must move together
if the objective is moved in focusing.
The significance of the arrangement lies in the fact that
the ray of light which traces the line BjC^ the limiting line for
the reading angle, traverses the principal focus of the lens P,
and hence emerges from this lens along the horizontal line,
nBy on which the cross-wire is placed. The lenses O and P are
so placed that the point Fp, which is the principal focus of Py
is also the focus of O, which is conjugate with the point C, the
center of the instrument. Any further discussion of the
theory of this instrument is out of place here until it is manu-
factured and used in this country.*
THE INSTRUMENTS.
206. The Transit. — That the transit may be best adapted
to this work, there are certain features it should possess,
though all of them are by no means essential. They will be
named in the order of their importance.
1st. The horizontal limb should be graduated from zero to
360°, preferably in the direction of the movement of the hands
of a watch.
* For the mathematical discussion of this telescope see an article by the authof
in Engimcfing News, November 8, i8go. Digitized byCjOOQlc
252 SURVEYING.
2d. The instrument should have a vertical circle rigidly at*
tached to the telescope axis, and not simply an arm that is
fastened by a clamp-screw, and which reads on a fixed arc be-
low. So much depends on the vertical circle holding its adjust
ment that its arrangement should be the best possible. Since
the telescope is not transited, the vertical circle need not 0e
complete.
3d. The telescope should be inverting, for two reasons :
first, in order to dispense with two of the lenses, and so obtain
a better definition of image ; and, second, that the objective
may have a longer focal length, thus giving a flatter image and
a less distorted field.
4th. The stadia wires should be fixed instead of adjustable,
as in the latter case they are not stable enough to be reliable.
5th. The bubbles on the plate of the instrument should be
rather delicate, so that a slight change in level may become
apparent. They should also hold their adjustments well. This
is very important, in order that the readings of the vertical
angles may be reliable. It is also of great importance in
carrying azimuth where the stations are not on the same level
6th. The horizontal circle should read to thirty seconds ;
and there should be no eccentricity, so that one vernier-read
ing shall be practically as good as two.
7th. The instrument (or tripod) should have an adjustable
centre, for convenience of setting over points.
8th. A solar attachment to the telescope will be found veiy
convenient. In most regions the azimuth can be checked up
by the reading of the needle, but in many places this is not
reliable.
207. Setting the Cross-wires. — The engineer should al-
ways have at hand a spider's cocoon of good wires, and a small
bottle of thick shellac varnish. If the dry shellac is carried it
may be dissolved in alcohol. If no such cocoon is at hand a
spider may be caught and made to spin a web. The small,
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TOPOGKAPHICAL SURVEYING. 253
black, outdoor spider makes a good web for stadia purposes.
A new wire should be allowed to dry for a few minutes, and an
old one should be steamed to make it more elastic. The
wires for stadia-work should be small, round, and opaque.
Some wires are translucent, and some are flat and twisted like
an auger-shank.
Scratches must be made across the face of the reticule
where the wires are to lie. These must be made with great
care, so as to have them equally spaced from the middle wire,
parallel to each other, and perpendicular to the vertical wire.
The distance apart of the extreme wires is to be computed by
equation (5) for any desired scale on the rod.
Take a piece of web on the points of a pair of dividers, by
wrapping the ends several times about the points, which should
be separated by about an inch ; stretch the wire, by spreading
the dividers, as much as it will bear; and lay the dividers
across the reticule in such a way that the web comes in place.
The dividers must be supported underneath, so that the points
will drop just a trifle below the top of the reticule; otherwise
they would break the web. Move the dividers until the web
is seen, by the aid of a magnifying-glass (the eye-piece will do),
to be in exact position. Then take a little shellac on the end
of a small stick or brush, and touch the reticule over the web,
being careful to have no lateral motion in the movement.
The shellac will harden in a few minutes, when the dividers
may be removed. Shellac is not soluble in water.
.208. Graduating the Stadia-rod. — The stadia-rod is
usually a board one inch thick, four or five inches wide, and
twelve to fourteen feet long. Sometimes this is stiffened by a
piece on the back. To graduate the rod, it is necessary to
know what space on the rod corresponds to a hundred feet (or
yards, or metres) in distance. Either of the three methods
cited on pp. 230-1 may be used for doing this, but the first is
recommended. Thus, measure off c +/in front of the plumb
'7
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254 SURVEYING,
line, and set a point. From this point measure off any con-
venient base, as 200 yards, on level ground, and hold the blank
rod (which has had at least two coats of white paint), at the end
of this base-line. Have a fixed mark or target on the upper
part of the rod, on which the upper wire is set. Have an assist-
ant record the position of the lower wire as he is directed by
the observer. Some sort of an open target is good for this pur-
pose, but any scheme is sufficient that will enable the observer
to fix theposition of the extreme wires at the same moment with
exactness. This work should be done when there is no wind,
and when the atmosphere is very steady : a calm, cloudy day
is best. Repeat the operation until the number of results, or
their accordance, shows that the mean will give a good result.
If the base was 200 yards long, divide this space into two equal
parts, then each of these parts into ten smaller parts, and
finally each small space into five equal parts; and one of
these last divisions represents two yards in distance. Dia-
grams are then to be constructed on this scale, in such a way
that the number of symbols can be readily estimated at the
greatest distance at which the rod is to be read. The individ-
ual symbols should be at least three inches across ; so that, if
one of these is to represent ten units, as yards or metres, then
100 units will cover 2^ feet, and a rod 14 feet long will read a
distance of 560 units (yards or metres). If it is desired to read
distances of a quarter of a mile or more, the rod should be
graduated to read to yards (or five-foot units, or metres); but
if it is not to be used for distances over 500 to lOOO feet, it
might be graduated to read to feet. This question must be
decided before the wires are set, and then they must be spaced
accordingly.
In measuring the base, care should be taken to test the
chain or tape carefully by some standard.
If the rod is to be graduated to read to feet, of course
the base should be some even hundreds of feet, as 60a
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TOPOGRAPHICAL SURVEYING.
255
In Fig. 63 are shown four designs for stadia-rods which
have been long in use, and are found to work well. They are
intended to be all in black on a white ground.* It will be
noticed that the shortest lines in these diagrams all cover a
space of two units on the rod. In diagrams 2 and 3 the units
are either yards or metres, while in i they are units of five
feet each. In diagram 4 the units are of two feet each. The
K ^ ^
»)0
-x^
-X'
^i^
*m
sod
3U0
Fia 63,
fed"
successive units are found at the middles and limits of these
lines and spaces. Wherever the wire falls, there should be a
white ground on some part of the cross-section ; and the more
white ground the better, provided the figures are distinct.
The black paint may be put on heavy, so that one coat will be
sufficient.
The 50- and loounit marks should be distinguished by
special designs. There should usually be at least two boards
with each instrument, and sometimes three and four are needed.
Of course, these are all duplicates. After the unit scale is
obtained, or the space on the rod corresponding to a hundred
* Some engineers prefer red on the loo-unit figures.
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256 SURVEYING,
units in distance, these loo-unit spaces should be so distributed
as to be symmetrical with reference to the ends of the rod. The
reason of this will appear later. Having determined how many
loounit spaces there will be on the rod, fix the position of the
two end ico-unit symbols with reference to this symmetry, and
then the rod is subdivided from these points.
Special pains should be taken to have the angular points of
the diagrams well defined and in position. These points are
on the lines of subdivision of the rod.
After one rod is subdivided, the others of that set may be
laid alongside, and all fastened rigidly together ; and then, by
means of a try-square or T-square, the remaining rods may be
marked off.
The wire interval should be tested every few months by
remeasuring a base, as was done for graduation, and reading
the rod on it, to see if this shows the true measured distance.
This is to provide against a possible change in the value of the
wire interval. If the wires are stretched reasonably tight when
they are' put in, they seldom change, If they are too loose,
they swell in wet weather, and may sag some. The reticule
should be so firm that the variable strain on the adjusting-
screws will not distort it appreciably.
If the wire interval is found to have changed, either the
rods must be regraduated, or else a correction must be made
to all readings of importance. What are called the "side
shots," which make up a large proportion of the readings
taken, would not need to be corrected.
If the wires are adjustable, any unit scale may be chosen
at pleasure, and the wires adjusted to this scale. Then, if the
intervals change, the matter is corrected by adjusting the
wires. The adjustable wires are generally used to obtain dis-
tances from levelling-rods, where it is desirable that each foot
on the rod shall correspond to a hundred feet in distance. For
the ordinary stadia-rods, fixed wires are preferable.
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TOPOGRAPHICAL SURVEYING, 257
GENERAL TOPOGRAPHICAL SURVEYING.*
209, The Topography of a region includes not only the
character and geographical distribution of the surface-cover-
ing, but also the exact configuration of that surface with
reference to its elevations and depressions. Thus any point
is geographically located when its position with reference to
any chosen point and a meridian through it is found, but to
be topographically located its elevation above a chosen level
surface must also be known. A topographical survey consists
in locating by means of three coordinates a sufficient number
of points to enable the intervening surface to be known or
inferred from these. Evidently the points chosen should be
such as would give the greatest amount of information. As
for geographical outline, the corners, turns, or other critical
points are chosen, so for configuration the points of change;
in slope, as the tops of ridges and bottoms of ravines, or the
brow and foot of a hill, are chosen as giving the greatest
information.
210. Field-work. — Let it be required to make a topo-
graphical survey of either a small tract, a continuous shore-
line, or of a large area, for the purpose of making a contour
map.
In case of the small tract, any point may be taken as a
point of reference, and the survey referred to it as an origin.
In case of an extended region, a series of points should br
determined with reference to each other, both in geographical
position and in elevation. These determined points should
not be more than about three miles apart. The points of ele-
vation or bench-marks need not be identical with those fixed
in geographical position. These last are best determined by a
system of triangulation, and are called " triangulation stations."
* See Appendix F for field methods used on the Mississippi Kiver Survey,
and foot-note, p. 662. For a complete treatise on Photographic Surveying, by E.
Deville, Surveyor General of Public Lands for the Dominion of Canada, apply lo
the Government Printing Bureau, Ottawa, Canada. The book contains 232 pages
and many cuts. Digitized by CjOOglC
258 SURVEYING.
First, a system of triangulation points is established, the
angles observed, azimuths and distances computed, and the
stations plotted to scale on the sheet which is to contain the
map. This plotting is best done, for small areas, by comput
ing the rectangular coordinates (latitudes and departures),
and plotting them from fixed lines which have been drawn
upon the map, accurately dividing it into squares of 1000 or
5000 units on a side. They may, however, be plotted directly
from the polar coordinates (azimuth and distance) as given by
the triangulation reduction. For this purpose, the sheet on
which the map is first drawn, called t\iQ field sheet , should have
a protractor circle printed upon ity about twelve inches in diam-
eter. ThtsQ protractor sheets of drawing-paper can be obtained
of most dealers in drawing-materials, or the protractor circle
may be printed to order on any given size or quality of paper.*
These protractor circles are very accurate, and are graduated
to 1 5' of arc. Plotting can be done to about the nearest 5'.
Second, a line of levels is run, leaving B.M.'s at convenient
points whose elevation are computed, all referred to a com-
mon datum. If the A's are not also B.M.'s, then a B.M.
should be left in the near vicinity of each A. This is not
essential, however.
Third, the topographical survey is then made, and referred
to, or hung upon, this skeleton system of A's and B.M.'s.
The topographical party should consist of the observer, a
recorder, two or three stadia-men, and as many axemen as
may be necessary, generally not more than two.
The azimuth, preferably referred to the true meridian, is
known for every line joining two A*s, as well as the length of
such line.
Set up the transit over a A, and set the horizontal circle
* Messrs. Queen & Co. Philadelphia, or Blattner & Adam of St Louis, can
furnish such sheets.
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TOPOGRAPHICAL SURVEYING, 259
(which should be graduated continuously from 0° to 360^ in
the direction of the hands of a watch) so that vernier A will
read the same as the azimuth of the triangulation line by which
the instrument is to be oriented. Clamp the plates in this
position, and set the telescope to read on the distant A. Now
clamp the instrument below, so as to fix the horizontal limb,
and unclamp above. The azimuths of the triangulation lines
are generally referred to the south point as the zero, and in
small systems of this sort the forward and back azimuths are
taken to be 180° apart. When the instrument has been set
and clamped, all subsequent readings taken at that station are
given in azimuth by the readings of vernier A on the horizon-
tal limb. For any pointing, therefore, the reading of this
vernier gives the azimuth of the point referred to the true
meridian, and the rod reading gives the distance of the point
from the instrument station. These enable the point to be
plotted on the map. To draw the contour lines, elevations
must also be known.
If the elevation of the A is known, measure the height of
instrument (centre of telescope) above the A on the stadia,* as
soon as the instrument is levelled up over that station. Sup-
pose this comes to the 212-unit mark. Write in the note-book,
as a part of the general heading for that station, ** Ht. of Inst.
= 212.** Then, for all readings from that station for eleva-
tions, bring the middle horizontal wire to the 212-unit mark
on the rod, and read the vertical angle. From this inclination
and distance, the height of the point above or below the
instrument station is found. If the rod be graduated sym-
metrically with reference to the two ends, one need not be
careful always to keep the same end down, and so errors from
this cause are avoided.
*0r, if preferred, a light staff, about five feet long, may be carried with the
instrument for this purpose, it bemg graduated the same as the stadia rods for
this instrument
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26o SURVEYING,
The record in the note-book consists of —
1st. A Description of the Pointy as, " N.E. cor. of house,"
"intersec. of roads,*' "top of bank," '*C.P." for " contour
point," which is taken only to assist in drawing the contours,
* El i6" for "stadia station i6," etc.
2d. Reading of Ver. A.
3d. Distance,
4th. Vert. Angle.
These four columns are all that are used in the field.
There should be two additional columns on the left-hand page,
for reductions, viz. :
5th. Difference of elevation, corresponding to the given
vertical angle and distance, and which is taken from a table or
diagram.
6th. Elevation, which is the true elevation of each point
referred to the common datum.
The right-hand page should be reserved for sketching.
It will be found most convenient to let the sketching pro-
ceed from the bottom to the top of the page ; as in this case
the recorder can have his book properly oriented as he holds
it open before him, and looks forward along the line. The
notes may advance from top to bottom, or vice versa, as de-
sired. If there are many "side shots" from each instrument
station, one page will not usually contain the notes for more
than two stations, and sometimes not even for one.
The sketch is simply to aid the engineer when he comes to
plot the work, and may often be omitted altogether. One
soon becomes accustomed to impressing the characteristics of
a landscape on his memory so as to be able to interpret his
notes almost as well as though he had made elaborate sketches.
For beginners the sketches should be made with care. The
observer should usually make his own sketches and plot his
own work.
After the instrument is oriented over a station, and its
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TOPOGRAPHICAL SURVEYING, 261
height taken on the stadia, the stadia-men go about holding
the rods at all points which are to be plotted on the map,
either in position or in elevation, or both. The choice of points
depends altogether on the character of the survey ; but since
a single holding of the rod gives the three coordinates of any
point within a radius of a quarter of a mile, it is evident the
method is complete, and that all necessary information can
thus be obtained. For very long sights, the partial wire inter-
vals (intervals between an extreme and the middle wire) may
be read separately on the stadia, and in this way twice as great
a distance read as the rod was designed for. The limit of
good reading is, however, usually determined by the state of
the atmosphere, rather than by the length of the rod. When
the air is very tremulous, good readings cannot be made over
distances greater than 500 feet ; while, when the atmosphere
is very steady, a half-mile may be read with equal facility.
Before the instrument is removed from the first station,
the forward stadia-man selects a suitable site for the next
instrument station (generally called stadia station, and marked
0, to distinguish it from a triangulation station, A), and drives
a peg or hub at this point. This peg is to be marked in red
chalk, with its proper number, and should have a taller mark-
ing-stake driven by the side of it. The peg for the (Z) should
be large enough to be stable ; for it must serve as a reference
point, both in position and elevation, during the period of the
survey. It is often desirable to start a branch line, or to
duplicate some portion of the work, with one of these stations
as the starting-point; and, since each El is determined, in
position and elevation, with reference to all the others, one
can start a branch line from one of these as readily as from a
A. It is not usually necessary to put a tack in the top, but
the centre may be taken as the point of reference. The stadia-
man first holds his stadia carefully over the centre of this B,
with its edge towards the instrument, so as to enable the
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262 SURVEYING.
observer to get a more accurate setting for azimuth. The
observer could just as well bisect the face of the rod ; but, if
held in this position, the centre of the rod may not be so
nearly over the centre of the peg as when held edgewise.
This holding of the rod edgewise for azimuth checks the care*
lessness of the stadia-man, and is done only for readings on
instrument stations.
At a signal from the observer, the stadia is turned with its
face to the instrument, and the observer reads the distance and
vertical angle.
It is advisable, in good work, to re-orient and relevel the
instrument just before reading to the forward EL The transit
is very apt to get out of level after being used for some time,
with more or less stepping around it, and the limb may have
shifted slightly on the axis, both of which might be so slight
as to make no material difference for the side readings, but
which would be important in the continued line itself. It is
best, therefore, to level up again, and reset on the back station,
before reading to the forward one. If it is inconvenient for
the rear rodman to go back to this station to give a reading, a
visible mark should be left there, to enable the observer to
reset upon it for azimuth, as it is not necessary to read distance
and vertical angle again.
When the instrument is moved, it is set up over the new
station, and the new height of instrument determined and
recorded. The rear stadia-man is now holding his rod, edge-
wise, on the station just left ; and by this the observer orients
his instrument, making vernier A read i8o^ different from its
previous reading on this line. Clamping the plates at this
reading, .the telescope is turned upon the rod on the back sta-
tion, and the lower plate clamped for this position. The circle
is now oriented, so that, for a zero-reading of vernier A, the
telescope points south.
// will be noted t flat the telescope is never reversed in this work
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TOPOGRAPHICAL SURVEYING. 263
■ r
The distance and vertical angle should both be reread, on
this back reading, for a check. If the vertical circle is not in
exact adjustment, this second reading of the vertical angle will
show it, for the numerical value of the angle should be the
same, with the opposite sign. If they are not the same, then
the numerical mean of the two is the true angle of elevation,
and the difference between this and the real readings is the
index error of the vertical circle. This error may be corrected
in the reduction, or the vernier on the vertical circle may be
adjusted.
The second reading of the vertical angle on the stadia-
stakes is thus seen to furnish a constant check on the adjust-
ment of the vertical circle, and should therefore never be
neglected. If the circle is out of adjustment by a small
amount, as one minute or less, in ordinary work it would not
be necessary either to adjust it or to correct the readings on
side-shots, for the elevations of contour points are not required
with such extreme accuracy. The mean of the two readings
on stadia-stakes would still give the true difference of elevation
between them, so that there would be no continued error in
the work.
The work proceeds in this manner until the next A is
reached. In coming to this station, it is treated exactly as
though it were a new El; and the forward reading to it, and
the back reading from it, are identical with those of any two
consecutive H's. Having thus occupied the second A, and
having oriented the instrument by the last El. turn the tele-
scope upon some other A vvhose azimuth from this one is
known. The reading of vernier A for this pointing should be
this azimuth, and the difference between this reading and the
known azimuth of the line is the accumulated error in azimuth
due to carrying it over the stadia line. This error should not
exceed five minutes in the course of two or three miles in crood
work.
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26d SURVEYING,
The check in distance is to be found from plotting the line,
or from computing the coordinates of the single triangulation
line, and also of the meandered line, and comparing the re-
sults.
The elevations are checked by computing the elevation of
the new A from the stadia line, and comparing this with the
known elevation from the line of levels.
In case the elevations of the A's are not given, but only
certain B.M/s in their vicinity, then the check can be made
on these just the same. Thus, in starting, read the stadia on
the neighboring B.M., and from this vertical angle compute the
elevation of the A over which the instrument sets, and then
proceed as before. In a similar manner, the check for eleva-
tion at the end of the line may be made on a B.M. as well as
on the A.
A quick observer will keep two or three stadia-men busy
giving him points; so that in flat, open country, with long
sights, it may be advisable to have three or even four stadia-
men for each instrument. In hilly country more time will be
required in making the sketches, and hence fewer stadia-men
are required.
After the instrument is oriented at each new station, the
needle should be read as a check. To make this needle-read-
ing agree with the readings of the verniers on the horizontal
circle (the north end with vernier A, and the south end with
vernier B, for instance), graduate an annular paper disk the
size of the needle-circle, and figure it continuously from o^ to
360°, /;/ the reverse direction to that on the horizontal limb of
the instrument, and paste it on the graduated needle-circle in
such a position that the north end of the needle reads zero
when the telescope is pointing south. If the variation is 6°
east, this will bring the zero of the paper scale 6° east of south
on the needle-circle. This position of the paper circle is then
good within the region of this variation of the needle. When
the survey extends into a region where the variation is differ-
ent, the scale will have to be reset.
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TOPOGRAPHICAL SURVEYING, 265
With these conditions, when the instrument is oriented for
a zero-reading when the telescope is south, the reading of the
north end of the needle will always agree with the reading of
vernier A, and the south end with vernier B. It is so easy a
matter to let the needle down, and examine at each El to see if
this be so, that it well pays the trouble. No record need be
made of this reading, as it is only used to check large errors.
211. Reducing the Notes.* — The only reduction necessary
on the notes is to find the elevation of all the points taken, with
reference to the fixed datum, and sometimes to correct the
distance read on the rod for inclined sights. The difference of
elevation between the H and any point read to, as well as
the correction to the horizontal distance, can be taken from
Table V, as given on pp. 772-779. The method of using this
table has been explained at length on p. 248. A very accurate
and by far the most expeditious method of taking out these dif-
ferences of elevation is by means of the Colby Slide Rule shown
in Fig. 63^. This is a slide rule 50 inches long, graduated so as to
:M.dmd0^
Fig. 63a.
give differences of elevation for any distance and for any angle
up to nineteen degrees. In addition to giving the differences
of elevation in the same unit (and it is immaterial what unit)
of measure used in reading the distances, this slide rule will
also give with equal faciHty the differences of elevation in feet
when the distances are read in either meters or yards. Fig. 63^2
shows only about one-tenth of the slide rule, reduced one-half.
It is for sale by leading dealers, and by B. H. Colby, St. Louis,
Mo. After the differences of elevation are taken out, the final
elevations of the points are to be computed by adding algebra-
ically the difference of elevation to the elevation of e3. jqqI^
The following is a sample page with these reductions:
• See note on p. 28o</.
266
SURVEYING.
April 20, 1883.
At EI4. Ht. of Inst. = 87.
Gazzam, Observer,
Bailr, Recorder.
Elevation = 24'.94.
Object.
H3
Bridge
S. E. cor. of house
On road
Water-level, foot of hill.
H5
C.P
Azimuth.
Ver. A.
Distance.
Vert.
Au^^le.
Difference
of
Elevation.
Eleva-
tion
above
Datum.
yds.
328° 10'
199
— o** 10'
- r.56
—
127" 40'
70
+ o"32'
+ I'.Q
26'.8
142'* 35'
90
+ 0' 15'
+ I'.2
26'. I
180* 25'
114
+ 0^ i
+ o'.7
25'.6
230*' 15'
224
-0^57'
— io'.9
14.0
128'' 33' 30'
216
+ 0^55'
+io'.38
—
190'* 48'
210
fl- 2'
+ii'.4
36'.3
At El 5. Ht. of Inst. = -78. Mean = + io'.26. 35'.20.
H4
S. W. cor. of house
Edge of bank
S.E. cor. of R.R. station.
Railroad track
[36
308"' 33' 30'
215
- 0" 54'
— 10'. 13
43" 30'
104
+ 3" 3'
+16.0
332** 10'
98
+ i°57'
+10'. I
85^ 30'
158
+ 1- 2'
+ 8'.5
43- 55'
40
+ 2\«;3'
+ 6'.o
79" 30'
270
+ o« 9
+ 2'.I
79* 30'
200
— o** 2'
— o'.36
51.2
45'.3
43'. 7
41.2
37'. 3
At 0 6. Ht. of Inst. = 79. Mean = — o'.54. 34'.66.
Els
Cor. of house.
Top of hill . . .
Wagon road. .
El 8
C.P
E]7
259 30
277' 55'
87^ 25'
58^ 15'
40" 37'
41^45'
5^25'
200
112
1 98
+ 0" 4'
+ 3^*26'
+ 4' 48'
186^ +4'*2S'
— -fo 33
+ 4^41'
+ 0'* 12'
213
III
194
+ o'.72
+19-7
+49'. 3
+42'.9
+73'.53
+27'.o
+ 2'.04
54.4
84.0
77'.6
6r'.7
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TOPOGRAPHICAL SURVEYING, 267
^^ 4. .
It will be noted that the reading on EI 5 fro*m El 4 has a
distance of 216 yards, and a vertical angle of +0^ 55' J while
on the back reading, from El 5 to E] 4 the distance is 215 yards,
and the vertical angle — o*' 54'. The distance was probably
between 215 and 216 yards, and the vertical circle was prob-
ably slightly out of adjustment. The difference of elevation
is taken out for both cases, however, being respectively 10.38
feet and 10.13 feet. The mean of these is 10.26 feet, which
stands as a part of the general heading at Q 5. The true
elevation of El 5 is then found by adding 10.26 to 24.94,
giving 35.20 feet, which is also set down as part of the general
heading.
The elevations on the side-readings from this station can
now be taken out. These side-elevations are only used for
obtaining the contours, and hence are only taken out to tenths
of a foot. When the contours are ten feet apart or more,
these side-elevations need only be taken out to the nearest
foot. The elevations of the stadia stations should, however,
always be taken out to hundredths, to prevent an accumula-
tion of errors in the line.
The reduction for distance may be taken from Table V,
p. 772, as It cannot be found with the Colby slide-rule.
This correction need only be made as indicated on p. 249,
and it is to be always subtracted from the rod-reading.
Thus, in the reading on E] 8 from El 6, we have a reading of
216 yards, and a vertical angle of 6° 33'. The correction here
is 2.16 X 1.3 = 2.8 yards, as found from the table. Calling this
3 yards it is subtracted from the 216, leaving 213 yards as
the distance to be plotted. It is only the stadia-line distances
that need ever be corrected in this way, the corrections being
usually so small that it is not important on the side-shots.
It will be noted that two E]*s were set from El 6. This
was done because a branch-line was run from El 6 over the
bluffs. In order to make it unnecessary to occupy E] 6 again
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268 SURVEYING.
when the bfanch-Hne came to be run, El 8 was set while B 6
was occupied in the main-line work. When the branch-line
came to be run, the instrument was taken directly to H 8, and
oriented on H 6 by the readings previously taken from EI 6.
The right-hand page of the note-book, opposite the notes
given above, is occupied with a sketch of the locality, with the
El's marked on, the general direction of the contour lines, the
railroad, stream, houses, etc.*
^\ 212. Plotting the Stadia Line.— It is customary to first
plot the stadia stations alone, from one El to the next, to find
whether or not it checks within reasonable limits. This part
of the work should be done with extreme care, so that if it
does not check it cannot be attributed to the plotting. In
case it does not check within the desired limit, then the line of
investigation will be about as follows until the error is found :
1st. Replot the stadia line.
2d. Recompute and replot the triangulation line.
3d. By examining the discrepancy on the plot, try and
decide whether the error is in azimuth or distance, and, if
possible, where such error occurred, and its amount.
4th. Examine the note-book carefully, and see if there is any
evidence of error there.
5th. If there is a large probability that the error is of a
certain character, and that it occurred at a certain place, take
the instrument to that station, set it up, and redetermine the
azimuths or distances which seem to be in error.
6th. If there is no high probability of any certain errors to
be examined for in this way, then go back and run the line
over, taking readings on ElV only. If the elevations had been
found to check, the vertical angles may be omitted on this
duplicate line; and, on the other hand, if the plot came out all
right, but the elevations could not be made to check, then a
duplicate line must be run to determine this alone ; and in this
* These notes were taken from a field-book of a topographical survey A
Cr^ve Cceur Lake by the engineering students of Washington University.
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TOPOGRAPHICAL SURVEYING. 269
case the vertical angles between [Ifs are all that need be read.
In cases of this kind, it will be found a great help to have the
ETs so well marked that they can be readily found.
With reasonable care in reading and in the handling of the
instrument, it will never be necessary to duplicate a line entire,
for all readings between ETs are checked. The vertical angles
and distances are checked by reading them forward and back
over every stadia line; and the azimuth is checked by the
needle readings, and also when the second A is reached.
If, in the progress of the work, the readings on the back 0
for distance and vertical angle do not fairly agree with these
quantities as read from the previous station, the recorder
should note the fact : and the observer should then re-examine
these readings; and, if found to be right, the first readings,
taken from the other station, should be questioned, and the
mean not taken in the reduction.
For plotting the stadia lines a parallel ruler (moving on
rollers) is very desirable ; otherwise, triangles must be used.
The plotting is done by setting the parallel ruler or triangle
on the proper azimuth as found from the protractor printed on
the sheet, moving it parallel to itself to the station from which
the point is to be plotted, and drawing a pencil line in the right
direction. Then, with a triangular scale, — or, better, with a
pair of dividers and a scale of equal parts, — lay off the correct
distance on this line ; and this gives the point.
If the instrument was oriented in the field for a zero read-
ing for a south pointing, then the protractor on the sheet must
have its south point marked zero, and increase around to 360°
in the same direction in which the limb of the instrument in-
creases, preferably in the direction of the movement of the
hands of a watch.
213. Check Readings. — To enable the observer to locate
large errors in azimuth or distance, or both, it is a good prac-
tice to take azimuth readings to a common object from a series
of consecutive stations, if such be possible. If the plot does
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2/0 SURVEYING,
not close, go back and plot in these azimuths ; and if there has
been no error in azimuth or distance between H's, and no error
in reading the azimuths for these pointings, then all these lines
will meet in a common point on the plot. If all but one in-
termediate line meet at a point, then the error probably war
in reading the azimuth of this pointing alone. If several of
the first pointings intersect in a point, and the remaining point-
ings of the set taken to this object intersect in another point,
then it is highly probable that the error was in reading the
azimuth or distance of the line connecting these two sets of
[Il*s; and the relative position of the points of intersection
will enable the observer to decide whether the error was in
azimuth or distance, and about how much. If, in this way,
the error be located, the instrument can be taken to this point,
and the readings retaken.
214. Plotting the Side-readings. — Having plotted the
stadia line and made it check, the next step is to go back and
plot in the side-readings. For doing this, a much more rapid
method may be used than that described above.
Divide the sheet into squares by horizontal and vertical
lines spaced uniformly at from 1000 to 5cxx> units apart, ac-
cording to scale. These lines are to be used for orienting the
auxiliary protractor, and also to test the paper for stretch or
shrinkage.
The side-readings are now plotted by the aid of a paper
protractor, such as is shown in Fig. 64. This is made from a
regular field-protractor sheet. The graduated circle printed
on the sheet is used ; and this is some 12 inches in diameter,
and graduated to 15 minutes. The sheet is trimmed down to
near the graduated circle, and the edges divided, as shown in
the figure, to any convenient small scale.* This sheet is to be
* It is sometimes desirable to make the open space DFE rectangular and
graduate the sides of the space ABF instead of the outer edges. The pro*
tractor can then be used nearer the edge of the sheet.
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TOPOGRAPHICAL SURVEYING.
271
laid upon the plot, with its centre, C, coinciding with the El
It is oriented by bringing the corresponding spaces on opposite
edges ^9iyiSWi?5Wf\WW^ ^"7 one of the spaced lines on the
plot. ^[iil^^^fPfli^^ position parallel to that of the
protractor circle printed on the sheet, and an azimuth taken
from the one will agree with an azimuth taken from the
other. When this auxiliary protractor has been so centred
and oriented, let it be held in place by weights. Now the
part ADEB folds back, on the line AB, into the position indi-
cated by the dotted lines. The portion DEF is cut out en-
iliiiiliiiiliiiiliiiiliiiiliiiiliiiilniihmlii'iliiiifr
Fic. 64.
tirely, so that when the flap is turned back the space AFB
is left open. This space is to be large enough to include the
longest side-readings when plotted to scale ; that is, the radius,
CF, of the circle to the scale of the drawing must exceed the
longest readings. We now have a protractor circle about the
Q, with this station for its centre.
Take a triangular scale, select the side to be used in laying
oil the distances, and paste a piece of strong paper on the
lower side at the zero point. Make a needle-hole through this
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paper close to the edge, at the zero of the scale. Fasten a
needle through this hole into the point which marks the exact
position of the H. The scale can now swing freely around the
needle, on tlie auxiliary protractor; and its zero remains at
the centre of the station from which the points are to be
plotted.
To plot any point, swing the scale around to the proper
azimuth, and at the proper distance mark with the pencil the
position of the point. If this marks a feature of the land-
scape, it should be drawn in at once, before going farther ; and
if the elevation of the point will be needed in sketching the
contours, this should also be written in. For contour points,
the elevation is all that is put down.
In this manner the points can be plotted very rapidly. A
six-inch triangular scale, divided decimally, will be found best
for this.
If there is very much of this work to be done, it might be
found advisable to have a special scale constructed for the
(?
TiliM|iiM|iiii|iiii|iiii|iiii|iiiiiiiii|iiiiiiiiiiiinTmT
a
Fig. 65.
purpose. Fig. 65 is one form of such a scale drawn one-third
size, which would be found verj' convenient and cheap. It
should be graduated on a bevel edge, and to such a scale that
the units of distance used on the rod may be plotted to the
scale of the drawing. The small needle-hole, in line with the
graduated edge, should be only large enough to fit the
needle-point used, so that there would be no play. The rule
then turns on an accurate centre, which will not wear. Such
scales, six inches long, could be constructed very cheaply of
German silver by any instrument-maker.
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TOPOGRAPHICAL SURVEYING.
VI
A special form of protractor, shown in Fig. 66, has also
been used with great success in France and on the Mississippi
River surveys.*
It is essentially a semicircular protractor, provided with
Fig. 66.
*• Manufactured by Mahn & Co , St. Louis, Mo.
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274 SURVEYING,
a needle-pointed pivot at its centre, and having the straight
edge graduated so that distances can be measured off each
way from the pivot ; the angular deflection is given by the
graduated circle, reading from a point marked on the paper.
The bottom of the plate is flush with the bottom of the pro-
tractor, and the hole F is at the centre, and should be only
large enough to admit a fine needle. The screw D has a hole
drilled in its axis to admit the needle-point. It is also split,
so that when it is screwed down it will clamp the needle
firmly. If the latter is broken, it can readily be replaced by a
new one. In addition to the scale on the beveled edge, a
diagonal scale is also provided as shown. This instrument
combines all the requisites for rapid and accurate plotting of
points located by polar co-ordinates or by intersections.
In using this protractor the needle-point is placed at, say,
the first station, and pressed firmly down. A meridian line is
then decided upon, and a point is marked on it at the outer
edge of the protractor circle. This will be the initial point
from which the angles will be read. As azimuth is read
from the south around by the west, it is plain that the circle,
numbered as shown and revolved about the pivot till the
proper reading coincides with the meridian line, will give the
direction of the required point along the graduated diameter,
while from the latter the distance can be pricked off. A point
can be plotted in any direction without lifting the protractor
from its position.
In going to the second station it is not necessary to draw
a meridian line through it. The azimuth between the first
and second stakes being known, if the pivot be set at the lat-
ter, and the protractor revolved so that the straight edge coin-
cides with the line passing through the two stakes, then the
point on the circle corresponding to the azimuth of the line
will be a point on the meridian line. This point being marked
on the paper is the origin for the angles plotted from the
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TOPOGRAPHICAL SURVEYING, 2/5
second station, and it is evident that they will bear the proper
relations to the points plotted from the first station.
Other methods are employed for plotting the side shots,
such as solid half-circle protractors, of paper or horn, weighted
in position, with their centres over the station. This is ori-
ented on a meridian drawn through the point, and then all the
points plotted whose azimuth falls between o° and i8o°, when
the protractor is laid over on the other side, and the remaining
points plotted. In this case the ruler is laid across the pro-
tractor, with some even division at the station. This method
is more troublesome, less rapid, and defaces the drawing more,
than the other methods given above. The plotter should have
an assistant to read off to him from the note-book. When
all the elevations have been plotted, the contour lines are
sketched in.
The plotting should keep pace with the field-work as close-
ly as possible, being done at night and at other times when the
field-work is prevented or delayed. In difficult ground the
map could be carried into the field and the contours sketched
in on the ground. At least the stadia lines should be plotted
up and checked before the observer leaves the immediate local-
ity. Where the elevations are checked on B.M.'s, these checks
should be immediately worked out. This much, at least, could
be done each evening for that day's work.
215. Contour Lines. — In engineering drawings the config-
uration of the surface is represented by means of contour lines,
A contour line is the projection upon the plane of the paper of
the intersection of a horizontal, or rather level, plane with the
surface of the ground. These cutting level planes are taken,
five, ten, twenty, fifty, or one hundred feet apart vertically,
beginning with the datum-plane, which is usually taken below
any point in the surface of the region. Mean sea-level is the
universal world's datum which should always be used when
a reasonably accurate connection with the sea can be ob-
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2^6 SURVEYING,
tained.* Such contour lines are shown on Plate II. The proper
drawing of these contours requires some accurate knowledge of
the surface to be depicted, aside from the elevations of isolated
points plotted on the map. This knowledge may consist of a
vivid mental picture of the ground, derived from personal ob-
servation, or it may be gained from sketches made upon the
ground. Even with this knowledge the draughtsman must
keep vividly in mind the true geometrical significance of the
contour line, in order to properly depict the surface by this
means. The ability to draw the contour lines accurately on a
field-sheet is the severest test of a good topographer. They
are first sketched and adjusted in pencil and then may be
drawn in ink.
A few fundamental principles may be stated that will assist
the young engineer in mastering this art.
1. All points in one contour line have the same elevation
above the datum-plane.
2. Where ground is uniformly sloping the contours must
be equally spaced, and where it is a plane they are also straight
and parallel.
3. Contour lines never intersect or cross each other.
4. Every contour line must either close upon itself or ex-
tend continuously across the sheet, disappearing at the limits
of the drawing. It cannot have an end within these limits (an
apparent exception, though not really one, is the following).
5. No contour should ever be drawn directly across a
stream or ravine. The contour comes to the bank, turns up
stream, and disappears in the outer stream line. If the bed of
the stream, or ravine, ever rises above this plane, then the
contour crosses it ; but in the case of a stream the crossing is
never actually shown. In the case of a ravine the crossing is
shown, if points have been established in its bed.
6. Where a contour closes upon itself, the included area
* See in Chapter XIV., Precise Levelling, Art. 408.
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TOPOGRAPHICAL SURVEYING, 277
is either a hill-top or a depression without outlet. If the
latter, it would in general be a pond or lake. In other words,
such contours enclose either maximum or minimum points of
the surface.
7. If a higher elevation seems to be surrounded by lower
ones on the plot, it is probably a summit ; but if a lower eleva-
tion seems to be surrounded by higher ones, it is probably a
a ravine, or else an error ; otherwise it is a depression without
outlet, in which case there would probably be a pool of water
shown.
8. Contour lines cut all lines of steepest declivity, as well
as all ridge and valley lines, at right angles.
9. Maximum and minimum ridge and valley contours must
go in pairs; that is, no single lower contour line can intervene
between two higher ones, and no single higher contour line can
intervene between two lower ones.
10. Vertical sections, or profiles, corresponding to any line
across the map, straight or curved, can be constructed from a
contour map, and conversely a contour map may be drawn
from the profiles of a sufficient number of lines.
11. Each contour is designated by its height above the
datum-plane, as the fifty-foot contour, the sixty-foot contour,
etc. In flat country, where the contour lines are few and wide
apart, always put the number of the contour on the higher
side, otherwise it sometimes may be impossible to tell on which
side is the higher ground.
12. In taking surface-elevations for determining contour
lines, points should always be taken on the ridge and valley
lines, and at as many intermediate points as may be desirable.
There are two general systems of selecting these points. By
one system points are chosen approximately in lines or sec-
tions cutting the contours about at right angles, the critical
points being the tops and bottoms of slopes ; while by the
other system points are selected nearly in the same contour
line, — that is, on the same horizontal plane, — the critical points
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being the ridge and valley points, these being the points of
maximum and opposite curvature in the contour lines them-
selves. By the second method one or two principal contours
may be followed continuously, the points being taken as nearly
as may be on these contour lines. If such principal contours
are 50 feet apart, then when these are accurately drawn on the
map, any desired number of additional contours may be inter-
polated between the principal ones.
216. The Final Map.* — The field-sheets are drawn as de-
scribed above, in pencil, or partly in pencil and partly in ink,
or wholly in ink, according to the use to be made of them. If
they are simply to serve as the embodiment of the field-sur-
vey, to be used only for the construction of the final maps,
they are usually left in pencil, a six-H pencil being used. The
field-sheets are usually small, about 18x24 inches. The final
sheets may be of any desired size. Usually several field-sheets
are put on one final sheet, which will be worked up wholly in
ink, or color, the scale remaining the same. The work on the
field-sheet is then simply transferred to the final sheet by the
most convenient means available. Tracing-paper (not linen)
may be used. This is carefully tacked or weighted down over
the field-sheet, and the principal features, such as triangulation
stations, stream and contour lines, roads, buildings, fence lines,
etc., are traced in ink. The tracing-paper is then removed and
laid upon the final sheet, orienting it by making the triangula-
tion stations on the tracing coincide with the corresponding
stations on the final sheet, where they have been carefully
plotted from the triangulation reduction. All the matter on
the tracing may now be transferred to the paper beneath by
passing over the inked lines with a dull point, bearing down
hard enough to leave an impression on the paper below. If
preferred, the tracing may have its under surface covered with
plumbago (soft pencil-scrapings), after the tracing is made, and
then with a very gentle pressure of the tracing-point will leave
a Hght pencil line on the final sheet. In either case, when the
♦ See also Appendix G. digitized by LiOOgle
TOPOGRAPHICAL SURVEYING. 27^^
tracing is removed, these lines may be inked in on the final
sheet.
If the map is to be photo-lithographed it must be drawn
wholly in black, as given in Plates II. and III. If not, it is best
to use some color in its execution.* The water-lines may be
drawn in blue, and the contours in brown on arable land, and in
black on barren or rocky land. In this way the character of the
surface may be partly given. Where the slopes are very steep-
the contour lines become nearly coincident, but to further em-
phasize the uneven character of the ground, cross-hatching, or
hachures, may be employed on slopes greater than 45° from
the horizontal. All these conventional practices are illustrated
on Plate III., except the use of colors, this map having been
drawn for the purpose of being photo-lithographed. Plate II. is
a photo-lithograph copy of a student's map of the annual field
survey of the engineering students of Washington University.
217. Topographical Symbols are more or less conven-
tional, and for that reason given forms should be agreed upon.
The forms given in Plate III. were used on all the Mississippi
River surveys made under the Commission, and are recom-
mended as being elegant and fairly representative or natural.
Evidently the rice, cotton, sugar, and wild-cane symbols would
find no place in maps of higher latitudes. The cypress-tree
symbols may be used for pine to distinguish them from decid-
uous growth, and the sugar-cane symbol could be used for
corn if desired. It is not important to distinguish between
different kinds of cultivated crops, since these are apt to change
from year to year, but it is sometimes desirable to do so to
give a more varied and pleasing appearance to the map. The
grouping of the trees in a large forest is also varied simply for
the appearance, to prevent monotony. Colors are sometimes
used in place of pen-drawn symbols, but these are necessarily
so very conventional as to require a key to interpret them, and
besides it makes the map look cheap and unprofessional.
* See Plate IV.
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28o SURVEYING,
2i8. Accuracy of Stadia Measurements.— The attain-
able accuracy depends upon the care taken in graduating the
rod (or in determining the stadia constants), and upon the
ruggedness of the country surveyed. Results of surveys have
shown that a higher degree of accuracy is attainable in moder-
ately rugged country where the line of sight passes at a con-
siderable distance from the ground, thus avoiding the exces-
sive and irregular errors of diflferential refraction. In general
it may be said that under the most unfavorable circumstances
an accuracy of one in three hundred is easily attained, while
under favorable circumstances this accuracy may be increased
t6 one in two thousand or more.* The results obtained in the
U. S. Lake Survey are perhaps a fair average for various con-
ditions. On that survey the errors of closure of one hundred
and forty-one meandered lines was computed with a mean
result of one in six hundred and fifty. The lengths of sight
averaged from eight hundred to one thousand feet, with a
maximum length of two thousand feet. The official limit of
error was one in three hundred. The average length of lines
run was one and a half miles.
On the Mexican Boundary Survey the transit and stadia
method was used for taking topography over one thousand
seven hundred and fifty square miles, as well as in the measure-
ment of the entire boundary line. The conditions under which
this work was done were most unfavorable for accuracy, but
in a trial measurement of one hundred miles by the stadia
and the chain the former was found, by a comparison with
the true triangulated distance, to be far more accurate than
the chain.
The following table will show the degree of accuracy at-
*For a statement of the effect of length of sight upon the accuracy, see
Bulletin of the University of Wisconsin, Engineering Series, Vol. I, Nq^
5, page 127, 1895. Also Engineering News ^ Vol. XXXIII, page 364.
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TOPOGRAPHICAL SURVEYING,
2Zoa
tained on this survey both as regards errors in azimuth and in
linear measurement.
Number
of Lines.
Aggregate
Length of
Lines
in Metres.
Average
Length of
Courses
in Metres.
Average
Number of
Courses
per Line.
Average
Error
in Distance
on Closing
equals i in
Average
Azimuth Error
on closing
per Kilometre
of Line Run.
29
49
28
12
III823.2
280706.8
290633.9
143352.0
253 0
356.7
437.7
580.4
15.2
16. 1
23.7
20.6
553
782
817
786
I' 55"
I' 04"
0' 43"
o'3/'
118
826515.9
386.2
18. 1
752
0' 59.6
21 8a. Accuracy of Levels Run by the Stadia and
Transit. — The errors in carrying levels by means of the stadia
and vertical angles is a function of the average vertical angle
employed. This may be seen from the following tabulation
of field-work taken from the report of the Mexican Boundary
Survey, 189 1 -6,
Number
of Lines.
(Circuits.)
Aggregate
Length
of Lines.
Sum of the
Vertical
Components
of Courses.
Average
Vertical
Angle
of Lines.
Error in Eleva-
tion on Closing
per Kilometre
of Line run.
Error
in Distance
on Closing
equals x in
14
55
28
17
metres
182960
338132
186426
112025
metres
1262.4
II706.I
I2139.8
I 1930.6
0 /
0 24
1 59
3 43
6 05
feet
0.17
.37
.49
.59
metres
0.053
.III
.150
.181
.123
842
114
819543
37038.9
3 3
.40
From the above table it will be seen that on 14 circuits,
averaging 13,070 metres (8.2 miles), run in rolling country and
employing a small vertical angle, the closing error in elevation
was 0.17 feet per kilometre; also that the closing errors in-
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28o*
SURVEYING,
crease quite rapidly as the average vertical angle increases,
until, in the most rugged country requiring an average vertical
angle of 6° 5', 17 circuits, averaging 6590 metres (4.1 miles),
have a closing error of 0.59 feet per kilometre.
The relation of error to vertical angle, as deduced from the
512 miles of closed stadia lines comprising the above table, is
best shown by the graphic curve in Fig. 66^?.
0.1 ~oj 0.S 53 oi 0.6 0.1
ERROR OF CL06URE IN FEET PER KILOMETER
Fia66«.
A good example of the use of the transit and stadia method
in running levels in city topographic surveys is found in the
recent topographic survey of St. Louis, In this survey a
transit and stadia line was run over forty miles long. At
twenty-four points along this circuit the line checked on
triangulation points and precise-level bench-marks with the
results shown in the table on the following page.
The average length of the lines between check-points was
1.7 miles, and the average error for this distance was 0.24 of a
foot or 0.18 of a foot per mile of line.
It should be noted that while the total accumulated error
in elevation for the entire forty miles was but 0.64 of a foot,
at a point on the line distant 20 miles from the beginning the
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TOPOGRAPHICAL SURVEYING.
2&OC
RESULTS OF LEVELING BY THE STADIA METHOD,
as obtained on the St. Louis Topographical Survey.
Sution.
Azimuth
Error.
Accumulated
Error
in Elevation
in Feet.
Distance
in Miles from
the Starting-
point.
Error
in Horizontal
Measurement.
ErrorofClosure
in Elevation
between
Check-points.
Feet.
2547
o'' i' 20"
+ 0.42
2.0
+ 1
: 387
0.42
2803
0 50
+ .46
4.1
+ 1
: 619
.04
2777
2 20
+ .17
6.2
+1
:ii77
.29
1332
2 00
+ .09
7.8
+ 1
■ 1149
.08
1393
2 00
-f .50
9.2
+ 1
• 987
.41
774
3 33
+ .52
10.9
+ 1
2000
.02
400
8 13
+ .09
12.3
+ 1
1084
•43
389
7 06
+ .08
14.6
+ 1
1203
.01
1839
9 32
4- .25
16.3
t\
1025
.17
1871
9 52
— .01
1S.4
965
.26
2008
10 42
+ .14
20.5
+ 1
836
.15
2067
II 42
+ .37
22.4
+ 1
877
.23
41
12 12
+ .39
23.8
+ 1
961
.02
2292
II 42
+ .77
25.2
+ 1
1063
.38
2304
10 42
+ 1.37
27.0
+ 1
1 139
.60
1699
10 42
4- I.OO
• 28.9
+ 1
1484
•37
566
9 40
+ 1.23
30.3
+ 1-
1644
.23
1500
9 05
+ 1.03
31.8
+ !•
1724
.20
1488
10 00
+ 0.68
33.5
+ !•
2267
•35
937
12 35
+ 0-94
34-9
+ 1:
3291
.26
958
9 18
+ 0.98
36.2
+ 1:
3945
.04
III5
9 30
-fo.64
37.9
+ 1:
5174
•34
2476
8 20
+ 0.36
39.8
+ 1:
6420
.28
124
8 20
+ 0.64
40.4
+ I : 6332
.28
error in elevation was zero, while in seven miles more it was
over twice the error at the end of the circuit, thus emphasizing
the fact that the errors in such work tend to compensate.
The following data regarding the accuracy of stadia sur-
veys made under the Mississippi River Commission is given as
a fair sample of the results obtained under that organization.
In 1896, thirty-six stadia circuits, with an average length of
two thousand seven hundred and fifty metres (1.7 miles), were
run over a certain rough, hilly country, between Dubuque, la.,
and Prairie du Chien, Wis., ranging in elevation from one hun-
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2Sod SURVEYING.
dred to three hundred feet, with an average erroi^in elevation
of 0.77 foot per circuit or 0.59 foot per mile of line run. The
maximum error in this work was 1.50 feet on a circuit 4000
metres (2.5 miles) long, and the minimum error in elevation
was 0.0 foot on a circuit 2 i(X) metres (1.4 miles) long. Seventy-
four other circuits, averaging twenty-nine hundred metres (1.8
miles), were, run in a nearly level river-bottom, with a resulting
error in elevation of 0.43 foot per circuit or 0.31 foot per mile
of line run. The maximum error in this work was 2.5 feet on
a circuit 4300 metres (2.7 miles) long, while the minimum
error in elevation on the same work was 0.03 foot on a circuit
4400 metres long. As indicating the general law of compen-
sating errors, it may be said that of the above 74 circuits 38
gave too high and 29 circuits too low elevations, while on
seven circuits the error was zero.
It should be noted that after such closing errors have been
properly adjusted among the various points of the circuity which
is the usual practice, the probable error in elevation of any
such adjusted values is very much smaller than the closing
error of the circuit, and for all mapping purposes it is far
within the limits of accuracy required. It may then be con-
cluded that for all kinds of accurate topographic work, with
the possible exception of special very large scale surveys, the
transit and stadia method is by far the best and most econom-
ical method to use.
Note. — For reducing stadia readings for ** difference of elevation ** and also
for '* correction for horizontal distance" an excellent diagram has been put on
the market by A. H. Abbott & Co. of Chicago, Hi. This was devised by Mr.
Morris K. Trumbull and is described by him in tht Journal of the Western
Society of Engineers ^ Vol. Ill, p. 1399. This diagram is 24 in. by 30 in. in size,
has blue lines on a white ground, on heavy paper, and gives angles up to 8*1
distances to 1600 feet, and elevations to 100 feet. It also shows the " correction
for horizontal distance " at a glance, while taking out the "difference of eleva-
tion." This is the most convenient diagram the author has ever seen. It is
said to give results as rapidly as the Colby slide-rule. The price is $1.00
post-paid.
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CHAPTER IX.
RAILROAD TOPOGRAPHICAL SURVEYING,
WITH THE TRANSIT AND STADIA.*
219. Objects of the Survey. — Since the transit and stadia
are the best means of making a general topographical survey,
so they are the means that are best adapted to make a prelimi-
nary railroad survey, so far as this is a topographical survey.
The map of a railroad survey may serve two purposes :
First, to enable the engineer to make a better location of
the Hne than could be done in the field.
Second, to give all necessary data relating to right of way,
as the drawing of deeds, assessment of damages, etc.
In flat or gently undulating country, it is not advisable to
locate by a map ; but even here the map is quite as essential
for determining questions relating to the right of way.
In either case, therefore, a good topographical map of the
line is of prime importance, and all the data for this map may
be taken on the preliminary survey. f
Both these ends may be served by the same map. The
method of location by contours (sometimes called " paper lo-
cation") is often absolutely necessary in rough ground, but is
still more often judicious in simpler work, inasmuch as a better
location can often be made in this way.
220. The Field-work. — In this case there would be no
A's or B.M.'s to check on; but the errors in distance and ele-
vation would be no more, probably, than are noVv made on
* The methods described in this chapter were novel when this work first
appeared in 1886, but have now (1901) been adopted by many railway locating
engineers. See an excellent article advocating the method in Engineering News ^
Vol. XLV, Feb. 21, 1901.
t By " preliminary survey " is here meant a survey of a belt of country which
it is expected will embrace the final line, and not a mere reconnoissance made
to determine the feasibility of a line, or which of several lines is the best.
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282 SURVEYING.
preliminary surveys. In fact, the errors in distance would not
be nearly so great, unless the chain be tested frequently for
length, and the greatest care taken on irregular ground. If a
chain ico feet long has 6oo wearing-surfaces, which most of
them have, and if each of these surfaces be supposed to wear
ooi inch, which it will do in the course of a 200- or 3(X)-mile
survey, then the chain has lengthened by six inches, or the
error in distance is now' i in 200 from this cause alone. If we
add to this the uncertain errors that come from chaining up
and down hill, and over obstructed ground, it is certain that
the stadia measures will be much the more accurate.
In the matter of elevations, since the local change of ele-
vation is alone significant, and not the total difference of ele-
vation of points at long distances apart, the line of levels
carried by the stadia would be amply sufficient for a prelimi-
nary survey.
The following observations are applicable to the prelimi-
nary survey for final location, when it is expected the line will
be included in the belt of country surveyed :
1st. All data should be taken that will contribute to the so-
lution of all questions of location, such as elevations for con-
tour Hnes ; streams requiring culverts, trestles, or bridges, and
the necessary size of each, if possible ; all depressions which
cross the line, and will require a water-way, together with the
approximate size of the area drained ; highways and private
roads or lanes ; buildings of all kinds, fences, and hedges ;
character of surface, as rock, clay, sand, etc. ; character of
vegetation, as cultivated, forest, prairie, marsh, etc. ; the loca-
tion of any natural rock that may be used for structures on the
line, such as culverts or abutments ; high-water marks if in a
bottom subject to overflow; and, in fact, all information which
will probably prove of value in determining the location, or in
making up a report with estimates to the board of directors, or
in letting contracts for earthwork.
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RAILROAD TOPOGRAPHY. ^83
2d. All data that may be found useful in respect to land
titles or right of way, or that may relate to claims for dam-
ages, such as section corners, boundaries, fences, buildings,
streets, roads, lanes, farm roads, cultivated and uncultivated
land, as well as such as may be cultivated, public and private
grounds, orchards, forests, together with the value of the forest
timber, mineral lands, stone quarries, proximity to villages,
etc. Since the bearings and position of all boundary-lines are
of great importance in the matter of right of way, every such
boundary should have at least two readings upon it in the
field ; and these should be as far apart as possible.
221. The Maps. — Before any plotting is done, two ques-
tions of importance must be decided. They are — firsts
whether one set of maps is to serve for both the location and
for the further use of the company, or whether a set of contour
maps, worked up in pencil, shall serve for the location, and
another set for the continuous use of the company; second^
what shall be the scale of the maps ? These will be argued
separately.
Whether one or two sets of maps will be decided on, will de-
pend largely on the care that is exercised with the locating-
sheets. If these are carefully worked up for the location, and
kept clean, they can be utilized for the final maps. If they
become too badly soiled by field use, new sheets would prob-
ably be substituted for the uses of the company.
If it is expected, at the start, to have a different set of
sheets for the final maps, then ** protractor sheets" should be
used for the location. In this case, plot on these sheets only
such of the field-notes as will contribute to the location ; and
these need only be plotted in pencil. When the location has
been made, such features may be transferred from the locating-
sheets to the final maps, as may be desired. These would con-
sist mainly in the stadia stations, the contours, and the located
line. The rest of the field-notes may then be plotted on the
final sheets, and the whole worked up in ink.
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284 SURVEYING,
If, on the other hand, one set of maps is to serve both pur-
poses, then it would, perhaps, be best to use plain sheets, as
the protractor circle would somewhat disfigure the final maps.
The protractor sheets would, however, furnish a ready means
of taking off the bearings of lines from the final charts, which
might be thought to compensate for the slight marring of the
map's appearance. If plain sheets are chosen, then they should
be divided into squares by lines drawn in ink parallel to the
sides of the paper, in the direction of the cardinal points of
the compass. Both the stadia stations and the side-readings
may then be plotted by means of the auxiliary protractor, this
being oriented by the meridian lines on the sheet. Even here,
only those readings would at first be plotted that will contrib-
ute to the location, and these marked in pencil. After the
location has been decided oa, and the location notes taken off,
as described below, then the stadia stations, contour lines, the
located line of road, and such other features as should be pre-
served on the final map, are inked in, and the map thoroughly
cleaned. The rest of the field-notes may now be plotted, and
the map finished up.
If the road runs through a settled region, the questions of
right of way are among the first things to be settled ; so that
preliminary maps showing the relation of the road belt to the
property lines are essential to the settlement of damages, and
to obtaining the right of way from the property-holders.
Coincident, therefore, with the making of maps to determine
the location must come the construction of preliminary right-
of-wa]^ maps or tracings. On these latter need be plotted only
the boundary-lines, fences, more important buildings, roads,
etc., or just sufficient to enable the right-of-way agent to nego-
tiate intelligibly with the property-owners.**^ Neither the lo-
* For an excellent article on the subject of right-of-way maps and permanent
railway-property records, by Charles Paine, see The Railroad Gazette of Nov.
14, 1884. Reprinted in book form in *' Elements of Railroading.''
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RAILROAD TOPOGRAPHY, 28$
eating nor the final map should be on a continuous roll. The
roll requires more room for storage, is more apt to get dusty,
and is much more inconvenient for reference. When sheets
are used, the survey plot covers a more or less narrow belt
across the map. One of the edges of the sheet, either where
the plot enters upon it or disappears from it, should be trimmed
straight, and the plot extended quite to this edge. This edge
is then made to coincide with one of the parallel or meridian
lines of the next sheet ; so that when the line is plotted, the
sheets may be tacked down in such a way as to show the con-
tinuous plot of the survey.
The scale of the map will depend on whether or not separate
sets of charts are to serve the purposes of location and of the
continuous use of the company. For the purpose of location,
a scale of 400 feet to one inch does very well ; but for the final
detail sheets the scale should be larger. If both purposes are
to be served by one set of maps, then the scale should be
about 200 feet to one inch,* with 5- or 10 foot contours. The
sheets should be about twenty by twenty-four inches.
222. Plotting the Survey.— In case the map is plotted on
a protractor sheet, the methods of plotting will be identical
with those for general topographical work, except that here
there will be no checks, either for distance, azimuth, or eleva-
tion, except such as are carried along or independently de-
termined. For distance, there is no check, except the dupli-
cate readings between instrument stations, unless the survey
is through a region which has already been surveyed. In this
case the section lines may serve as a check on the distances.
The azimuth should be checked at every station by reading
the needle, as described on p. 264, and also by independently
determining the meridian frequently, either by a solar attach-
ment or by a stellar observation. If the line is not nearly
* Some engineers prefer a scale of 100 feet to one inch for the final charts ol
the company.
19
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286 SURVEYING.
north and south, or, in other words, if it is extended materiall)^
in longitude, then the azimuth must be constantly corrected
for convergence of meridians, as is shown in Chap. XIV.
The elevations can only be checked by the duplicate read-
ings between instrument stations.* All the greater care
should be used, therefore, on readings between stations.
The first plottingy whether there are to be two sets of maps
or one, will consist in representing on the sheet only such data
as will assist in deciding on the location. These will be mainly
contour points, streams, important buildings near the Hne,
principal highways, other lines of railway, villages with their
streets and alleys near the proposed location, the lines of de-
markation between cultivated and timbered or wild land, etc.
From the plotted elevations, aided by the sketches in the note-
book, the contour lines are drawn in ; if necessary, this may
be done on the ground. This is sufficient for determining
upon a location.
When this has been done, then the natural features, the
contour lines, the stadia stations, and the located line, may be
inked in (or transferred by means of tracing-paper, in case the
final maps are to be on separate sheets), and the remainder of
the notes plotted.
In drawing the contour lines in ink, make those upon bar-
ren or rocky land in black, and those on arable land in brown.
If they are ten feet apart, make every tenth one very heavy, and
every fifth one somewhat heavier than the others. If this be done,
only the 50- and loofoot contours need be numbered. In case a
map does not contain at least two of these numbered contours,
then every contour which does appear on the map should be
numbered, giving its elevation above the datum of the survey.
* It may be observed that the same lack of sufficient checks on the distance,
azimuth, and elevation obtains with the ordinary preliminary survey with tran-
sit, level, and chain. If preferred, all bearings may be taken from the needle, and
then each alternate station only need be occupied by the instrument See series of
articles on this subject by the author in ** The Railroad Gazette " for Feb. 3d, Mai;
2d, 9th, and 30th, 1888,
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RAILROAD TOPOGRAPHY. 28/
The streams should be water-lyied in blue, and an arrow
should tell the direction of its flow. The name should also be
given when possible.
All fences should be shown, and especial pains taken to
represent division fences in their true position ; for it is from
this map that the deeds for the right of way are to be drawn.
Outhouses may be distinguished from dwellings by diago-
nal lines intersecting, and extending slightly beyond the out-
line. The character of the buildings may be shown by colors,
as red for brick, yellow for frame, pale sepia for stone ; the
outlines always being in black.
The stadia stations should be left on the finished sheets;
as, in case of a disputed boundary, or for other cause, the map
may be replotted if the positions of the instrument stations
are left on it. The numbers of the stations should, of course,
be appended.
The magnetic bearings of boundary-lines may be given on
the map, or they may be determined, as occasion requires, by
means of the auxiliary protractor and the true meridian lines
when the variation of the needle is known. For this purpose,
the magnetic meridian should be drawn on each map, diverg-
ing from one of the meridian lines, and the amount of the
Variation marked in degrees and minutes. •
223. Making the Location. — When a preliminary survey
is made, as above described, for the purpose of making what
is called a " paper location," the location is first made on the
map, and then staked out in the field.
Every railroad line is a combination of curves, tangents,
and grades; and it is the proper combination of these which
makes a good location. If it be assumed that the line is to be
included in the belt of country surveyed, then the map con-
tains all the data necessary to enable the engineer to select the
best arrangements of curves, tangents, and grades it is possible
for him to obtain on this ground. This selection can be made
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288 SURVEYING.
with much more certaint>^than is possible on the ground,
where the view is generally obstructed, and where grades are
so deceptive.
It is no part of this treatise to discuss the various problems
that enter into the question of a location, but only to show
how to proceed to make a location that may satisfy any given
set of conditions, by means of the contour map.
The contours themselves will enable the engineer to decide
what the approximate grades will have to be. Suppose a grade
of O-S foot in loo feet, or 26.4 feet to the mile, has been fixed
upon. It is now known that the line should follow the gene-
ral course of the contours, except that it should cross a lofoot
contour every 2000 feet. Spread the dividers to this distance,
taken to scale, and mark off in a rough way these 2000-foot
distances as far as this grade is to extend ; and do the same
for the successive grades along the line. Knowing the grade
of the line at the beginning of the sheet, the problem is to ex-
tend this line over the sheet so as to give the best location
one can hope to get on this ground with the available
means.
First, starting from the initial fixed point of line on the
map, sketch in a line which will follow the contours exactly,
crossing them, however, at such a rate as to give the necessary
grade. This is the cheapest line, so far as cut and fill are con-
cerned. Of course, where depressions or ridges are to be
crossed, the line must cross over from a given contour on one
side to the corresponding contour on the other, and then fol-
low along the contour again.
Second, mark out a series of tangents and curves which will
follow this sketched line as nearly as it is possible for a rail-
road to follow it. This will not be the final location, but it is
valuable for study. This line will be faulty from having too
many and too sharp curves, and too little tangent.
Third, draw in a third line, as straight as possible, and with
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RAILROAD TOPOGRAPHY,
289
as low grade of curves as possible cofisistent with a reasonable
amount of earthwork and a proper distribution of the same.
For the purpose of deciding what degree of curve is best
suited to the ground for a given deflection-angle, it is well to
have a series of paper templets made, with the various curves
for their outer and inner edges. Of course, these are cut with
radii laid off to the scale of the drawing. It is still more con-
venient to have these curves, laid off to scale, on a piece of
isinglass, horn, or tracing-paper (not linen), so that this can be
laid upon the map, and the curve at once selected which will
follow the contours most economically. Fig. 66 shows such a
series of curves drawn to a scale of i6cx> feet to the inch.
Fig. 66.
In this way the line is laid out over the map. The ques-
tions of greater or less curvature have been balanced against a
less or greater first cost, and greater or less operating expense.
The question of shifting it laterally has also been examined,
and finally a definite location fixed upon which seems to answer
best to the case in hand. When this is done, it only remains
to make up the location notes from which the line is to be
staked out.
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SURVEYING.
The following is considered a good form for the location
notes :
Location Notes for ABC Railroad. From Map No
Line.
Azimuth and
Deflection
Angles.
Lenjfth.
Station.
Remarks.
ft.
T
260" 40'
1020
10+ 20
P.C.
3* C.R.
+ 18^ 30'
617
16+37
P.T.
T
279° 10'
2670
43+ 7
P.C.
4^ C.L.
— 12** 20'
308
46+15 j
P.T.S. 46' 30' W.
0 12 320 ft
T
266" 50'
680
52 + 95
P.C.
The first column designates the tangents and curves, and
gives the degree of the curve, and the direction of its curva-
ture, whether right or left. If it curve toward the right, the
azimuth of the next tangent will be increased, and hence its
sign is plus, and vice versa.
The second column gives the azimuths of the tangents and
the deflection-angles of the curves. Each azimuth is seen to
be the algebraic sum of the two preceding angles.
The third column gives the lengths of the tangents as meas-
used from the map, and the lengths of the curves as determined
by dividing the deflection-angle by the degree of the curve.
Thus, 12° 20'= I2°.33, and 12^33-^4=308, which is the
length of the curve in feet.*
The fourth column gives the stations and pluses for the
P.C.*s and the P.T.'s. These quantities are simply the con-
tinued sum of those in the third column.
The first, second, and fourth columns now give all the infor-
* It is a great convenience to have at least one vernier, in railroad work,
gfraduated to read to hundredths of a degree. The case here given is only one
of many similar cases; but the principal advantage is in running the fractional
parts of curves when the curve chosen is some even degree, as here taken.
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RAILROAD TOPOGRAPHY. 2^\
mation necessary to stake out the line. The stadia is no longei
to be used, but a transit and chain, as is ordinarily done.
The tangents need not be run out to their intersection ; but
when the P.C. is reached, according to the location notes taken
from the map, set up the instrument, and stake out the curve
as far as possible, or around to the P.T. In either case, when
the instrument is to be moved, make a note of the forward
azimuth, and go forward and orient on the last station the
same as when moving between two G*s. If the instrument be
moved to the P.T. direct, then, after orienting back on the
P.C, turn off to the azimuth given for the next tangent, and
go ahead. The tangents could be run out to the intersection
and the point occupied by the instrument, for a check, if
thought desirable. The telescope is never reversed in laying out
the line from the system of notes above given.
With careful work, the line ought thus to be run out, and
the curves put in at once. Wc have supposed there was no
regular line cleared out on the preliminary, so the necessary
clearing would all have to be done on the location.
A levelling party follows the transit, and obtains the data
for constructing a profile and for determining the exact grades.
The stadia has served its purpose when it has enabled the
engineer to select the most favorable position for the line.
The transit, chain, and level must do the remainder. It is not
improbable that occasional modifications will be introduced in
the field, even though the survey and the location have been
made with the greatest possible care.
224. Another Method of making the preliminary survey
from which to determine the final location is as follows:
Run a transit and chain line, setting loO-foot stakes, as
nearly on the line of the road as can be determined by eye.
Follow this party by a level party which obtains the profile of
the transit line. A third party of one or more topographers
takes cross-sections at each too foot stake by means of a
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^9^ SURVEYING.
pocket-compass, clinometer, and hanii-ievel. These cross-
sections show the ground on either side of the line as far as
desirable by slope and distance, these latter being either meas-
ured by tape or paced. It is evident that contour lines could
be worked out from these data, but these would not be needed
if the distances and slopes were well determined, since these
give a better cross-section than contours alone could do.
The objections to this method are in the poor means it fur-
nishes for accurate determination of either distances or slopes,
and the haste with which it is usually done. There can be no
question but that accurate distances and slopes on cross-sections
ICO feet apart would give fuller data than even five-foot con-
tours accurately drawn. But to be accurately determined the
slope would have to change at all points — in other words, it
would be a curve. As to whether the slopes and distances as
they would probably be taken would give a better idea of the
ground than five-foot contours determined by the stadia
method, and the relative cost o\ the two systems, are matters
of experience. Both systems are competent to give a good
location when they are well executed.
Note. — The further study of railroad surveying falls within the province of
the various railroad field-books, which are printed in pocket form ana contain
the necessary tables for laying out a line of road. Having learned the con.
struction and use of surveying instruments, and the general methods of topo-
graphical surveying and levelling, the special applications to railroad location
given in the field-books are readily mastered. They will therefore noi bv
further considered in this work.
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CHAPTER X.
HYDROGRAPHIC SURVEYING.
22$. Hydrographic Surveying includes all surveys, for
whatever purpose, which are made on, or are concerned with,
any body of still or running water. Some of the objects of such
surveys are the determination of depths for mapping and navi-
gation purposes ; the determination of areas of cross-sections,
the mean velocities of the water across such sections, and the
slope of the water surface ; the location of buoys, rocks, lights,
signals, etc. ; the location of channels, the directions and ve-
locities of currents, and the determination of the changes in
the same ; the determination of the quantity of sediment car-
ried in suspension, of the volume of the scour or fill on the
bottom, or of the material removed by artificial means, as by
dredging.
A hydrographic survey is usually connected with an ex-
tended body of water, as ocean coasts, harbors, lakes, or riv-
ers. The fixed points of reference for the survey are usually
on shore, but sometimer buoys are anchored off the shore and
used as points of referejce. All such points should be accu-
rately located by triangulation from some measured base
whose azimuth has been found. The buoys will swing at
their moorings within small circles, these being larger at low
tide than at high, but the errors in their positions should never
be sufficient to cause appreciable error in the plotted positions
of the soundings. Where soundings need to be located with
great exactness, buoys could not be relied on. The triangula-
tion work for the location of the fixed points of reference dif-
fers in no sense from that for a topographical survey. In fact,
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^94 SURVEYING,
a hydrographic survey is usually connected with a topographical
survey of the adjacent shores or banks, the triangulation
scheme serving both purposes. It is not uncommon, however,
to make a hydrographic survey for navigation purposes sim-
ply, wherein only the shore-line and certain very prominent
features of the adjacent land are located and plotted. This is
the practice of the U. S. Hydrographic Office in surveying for-
eign coasts and harbors. In this case the work consists almost
wholly in making and locating soundings for a certain limiting
depth, as one hundred fathoms, or one hundred feet, inward
to the shore, and along the coast as far as desired. The length
and azimuth of a base-line are determined and the latitude ob-
served by methods given in Chapter XIV. The longitude is
found by observing for local time, and comparing it with the
chronometer time which has been brought from some station
whose longitude was known. Whenever telegraphic com-
munication can be obtained with a place of known longitude,
the difference between the local times of the two places is
found by exchanging chronographic signals. No special de-
scription will be here given of the methods used in this part of
the work, as they are all fully described in Chapter XIV.
THE LOCATION OF SOUNDINGS.*
226. Methods. — The location of a sounding can be found
with reference to visible known points by (i) two angles read
at fixed points on shore ; (2) by two angles read in the boat;
(3) by taking the sounding on a certain range, or known line,
and reading one angle either on shore or in the boat ; (4)
by sounding along a known range, or line, taking the soundings
at known intervals of time, and rowing at a uniform rate ; (5)
by taking the soundings at the intersections of fixed range
lines ; (6) by means of cords or wires stretched between fixed
*See Appendix F, and foot-note p. 662.
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HVDROGRAPHIC SURVEYING, 295
stations, these having tags, or marks, where the soundings are
to be taken. These methods are severally adapted to differ-
ent conditions and objects, and will be described in order.
227. Two Angles read on Shore. — If two instruments
(transits or sextants) be placed at two known points on shore,
and the angles subtended by some other fixed point, and the boat
be read by both instruments, when a sounding is taken, the in-
tersection of the two pointings to the boat, when plotted on the
chart containing the points of observation duly plotted, will
be the plotted position of the sounding. If three instruments
are read from as many known stations, then the three point-
ings to the boat should intersect in a point when plotted, thus
furnishing a check on the observations. The objections to
this method are that it requires at least two observers, and
these must be transferred at intervals, as the work proceeds, in
order to maintain good intersections, or in order to see the
boat at all times. While an observer is shifting his posi-
tion the work must be suspended. If there are long lines of
off-shore soundings to be made and there are no fixed points or
stations on shore of sufficient distinctness or prominence to be
observed by the sextant from the boat, then this method must
be used. When the angles are read on shore signals should be
given preparatory to taking a sounding, and also when the
sounding is made. If, however, the soundings are taken at
regular intervals the preparatory signal may be omitted, and
only the signal given when the sounding is taken. This
usually consists in showing a flag. The instrument may be set
to read zero when pointing to the fixed station. This reading
need only be taken at intervals to test the stability of the
instrument.
228. By Two Angles read in the Boat to three points on
shore whose relative positions are known. This is called the
"three-point'* problem. Let A, C, and B be the three shore
points, being defined by the two distances a and b and the angle
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SURVEYING.
C. Let the two angles P and F be measured at the point P,
The problem is to find the distances^/^and BP,
(a) Analytical Solution. — Let the un-
known angle at A be jr, and that at B be
B y^ Then we may form two equations
from which x and y may be found.
For,
__ ^i sin jr __ ^ sin y
Also, X + y ^ T^ed" - {P-\- F '\- C) = R.
From (2), y = R — X,
and sin j' = sin i? cos x — cos R sin x.
Substitute this value of sin y in (i), reduce, and find
a s\r\ F -\- b sin P cos R
. . (I)
. . (2)
cot X =
b sin P sin R
_ / ^ sin P' , \
= cot^'T-^— 5 B+l). . .
\b sm P cos R ' /
(3)
When X and y are found, the sides AP and BP are readily
obtained. This is perhaps the simplest analytical solution of
the problem.
(b) Geometrical Solution. — The following geometrical solu-
tion is of some interest, though it is seldom used :
Let A, Cy and B be the fixed points as before, and /^and
F the observed angles. Having the points^, B, and C plotted
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HYDROGRAPHIC SURVEYING. 297
in their true relative positions, draw from A the line AD,
making with AB the angle P {CPB), and from B the line BD,
making with AB the angle P (APC),
cutting the former Hne in D. Through
A, D, and B pass a circle, and through C
and D draw a line cutting the circum-
ference again in P. The point P is the
plotted position of the point of observa-
tion from which the angles P and P were
measured.
For P must lie in the circumference
through ADB by construction, otherwise
ABD would not be equal to APD, as they *''^- ^
are both measured by the same arc AD, The same holds for
the angle P', Also, the line PD must pass through C, other-
wise the angle A PC would be greater or less than P, which
cannot be. The point P\s therefore on the line CZ?, and also
on the circumference of the circle through ADB, whence it is
at their intersection.
This demonstration is valuable as showing when this
method of location fails to locate, and when the location is
poor. For the nearer the point D comes to Cthe more un-
certain becomes the direction of the line CD, and when D falls
at C — that is, when P is on the circumference of a circle through
A,B, and C — the solution is impossible, inasmuch as P may
then be anywhere on that circumference without changing the
angles Pand P, This is also shown by equation (3), above;
for if A, C, B, and Pall fall on one circumference, then x -\- y
^= R=i, 180°; whence cot x= 00 X O, which is indeterminate.
For cot ^ = — 00, and cos ^ = — i. Also a sin P = 6 sin P,
both being equal to the perpendicular from C on AB, The
equation then becomes
cot X = oofi — i) = 00 X O.
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^9^ SURVEYING.
^
{c) Mechanical Solution, — If the three known stations be
plotted in position and the two observed angles be carefully
set on a three-armed protractor,* then when the three radial
edges coincide with the three stations, the centre of the pro-
tractor circle corresponds to the position of the point of obser-
vation. With a good protractor this method gives the posi-
tion of the point as closely as the nature of the observations
themselves would warrant. It is the common method of plot
ting soundings when two sextant angles have been read from
the sounding boat.
Wood's double sextant (see p. 1 13) is designed to read these
two angles simultaneously. In the hands of an expert observer
this instrument is very valuable for surveys on running water.
(df) Graphical Solution. — The angles may be laid off on
tracing-paper or linen by lines of indefinite length, and this
laid on the plot and shifted in position until the three radial
lines coincide with the three stations, when their intersection
marks the point of observation. This is ttie most ready
method of plotting such observations when no three-armed
protractor is available.
The advantages of this method of locating soundings are
that it requires but one observer, no time is lost in changing
stations, and the party are all together, and hence there -can be
no misunderstandings in regard to the work. If the soundings
are made in running water, so that the boat cannot be stopped
long enough to read two sextant angles, two sextants are
sometimes used with one observer, he setting both angles and
reading them afterwards ; or two observers may be employed
in the same boat and the angles taken simultaneously.,
229. By one Range and one Angle.— The range may be
two stations or poles set in line on shore, or it may consist of
one point on shore and a buoy set at the desired position off-
* For description, with cut, see p. 167.
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HYDkOGkAPHIC SUk y EYING,
299
shore. If buoys are used they must be located by triangulation
from the shore stations. A triangulation system along a rocky
or wooded coast may consist of one line of sta-
tions on shore and a corresponding line of buoys.
The angles are read only from the shore stations, ^->v^
two angles in each triangle being observed. If
the buoys are well set and the work done in cahn
weather, the results will be good enough for to-
pographical or hydrographical purposes. The
stations and buoys should be opposite each other,
as in the figure, and readings taken to the two
adjacent shore stations and to the three nearest
buoys from each shore station. If the length of
any line of this system be known, the rest can be
found when the angles at A, B, C, and D are
measured. In such a system the measured lines
should recur as often as possible, ordinary chain-
ing being sufficient.
230. Buoys, Buoy-flags, and Range-poles.— A conveni-
ent buoy for this purpose may be made of any light wood,
eighteen inches to three feet long in tidelcss waters, and long
enough to maintain an erect position in tide-waters. It should
be from six to ten inches in diameter at top, and taper towards^
the bottom. If the buoy is not too long, a hole may be bored
through its axis for the flag-pole, which may then project two
or three feet below the buoy and as high above it as desired.
The buoy rope is then attached to the bottom end of the pole
and made of such length as to maintain the pole in a vertical
position in all stages of the tide. The anchor may be any suffi-
ciently heavy body, as a rock or cast-iron disk. If the buoys
are liable to become confused on the records, different designs
may be used in the flags, as various combinations of red, white,
and blue, all good colors for this purpose.
The range-poles should be whitewashed so as to show up
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300 SURVEYING,
against the background of the shore. The ranges are desig-
nated by attaching to the rear range-poles slats (barrel-staves
would serve) arranged as Roman numerals when read up or
down the pole. If range-poles are relied on, they must be very
carefully located and plotted, in order to establish accurately
a long line of soundings from a very short fixed base.
The observed angle may be either from the boat or from a
point on shore. In either case any other range-post of the
series may be used either for the position of the observer, if
on shore, or for the other target-point if the angle is read from
the boat.
231. By one Range and Time-intervals. — This is a very
common and efficient method, and quite satisfactory where
soundings need not be located with the greatest accuracy and
where there is no current. A boat can be pulled in still water
with great uniformity of speed ; and if the soundings be taken
at known intervals with the ends of the line of soundings fixed,
the time-intervals will correspond almost exactly with the
space-intervals. If the ends of the line of soundings are not
fixed by buoys or sounding-stations on shore, but the line sim-
ply fixed by ranges back from the water*s edge, the positions of
the end soundings may be fixed by angle-readings and the bal-
ance interpolated from the time-intervals.
232. By means of Intersecting Ranges. — This method
is only adapted to the case where soundings are to be repeated
many times at the same places. When the object of the sur-
vey is to study the changes occurring as to scour or fill on the
bottom it is very essential that the successive soundings should
coincide in position, otherwise discrepant results would prove
nothing. Such surveys are common on navigable rivers and
in harbors. Many systems of such ranges could be described,
but the ingenious engineer will be able to devise a system
adapted to the case in hand.
233. By means of Cords or Wires.— In the case of a fixed
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HYDROGRAPHIC SURVEYING, 3OI
but narrow navigable channel, having an irregular bottom, or
undergoing improvement by dredging, it may be found advis-
able to set and locate stakes on opposite sides of the channel,
to stretch a graduated cord or wire between them, and to locate
the soundings by this. By such means the location would be
the most accurate possible.
MAKING THE SOUNDINGS.*
234. The Lead is usually made of lead, and should be long
and slender to diminish the resistance of the water. It should
weigh from five pounds for shallow, still water, to twenty
pounds for deep running water, as in large rivers. If depth
onl)' is required, the lead may be a simple cylinder something
like a sash-weight for windows. If specimens of the bottom
are to be brought to the surface at each sounding, the form
shown in Fig. 71 may be used to advantage. An iron stem,
/, is made with a cup, r, at its lower end. The
stem has spurs cut upon it, or cross-bars attached
to it, and on this is moulded the lead which gives
the requisite weight. Between the cup and the
lead is a leather cover sliding freely on the shank
and fitting tightly to the upper edges of the cup.
When the cup strikes the bottom, it sinks far
enough to obtain a specimen of the same, which
is then safely brought to the surface, the leather
cover protecting the contents of the cup from be-
ing washed out in raising the lead. A conical cav-
ity in the lower end of the lead, lined with tallow,
is often used, and it is found very efficient for in-
dicating sand and mud. It is often very essential to know
whether the bottom is composed of gravel, coarse or fine, sand,
mud, clay, hard-pan, or rock, and this knowledge can be ob-
tanied with the cup device described above.
235. The Line should be of a size suited to the weight of
• See Appendix F for a tlescriptioD of methods used io the Miss. River Survey.
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302 SURVEYING,
the lead, and made of Italian hemp. It is prepared for use by
first stretching it sufficiently to prevent further elongation in
use after it is graduated. Probably the best way to stretch a
line is to wind it tightly about a smooth-barked tree, securely
fasten both ends, wet it thoroughly, and leave it to dry. Then
rewind as before, taking up the slack from the first stretching,
and repeat the operation until the slack becomes inappreciable.
It may now be graduated and tagged. Sometimes it is fastened
to two trees and stretched by means of a ** Spanish windlass,**
and then wet. It is quite possible to stretch the line too much,
for sometimes sounding-lines have shortened in use after being
stretched by this method. Soundings at sea are taken in fath-
oms. On the U. S. Lake Survey all depths over twenty-
four feet (four fathoms) were given in fathoms, and all
depths less than that limit were given in feet. On river and
harbor surveys it is common to give depths in feet. Channel-
soundings on the Western rivers made by boatmen are given
in feet up to ten feet, then they are given in fathoms and quar-
ters, the calls being "quarter-less-twain,** " mark-twain,*' "quar-
ter-twain," ** half-twain,** ** quarter-less-three," " mark-three."
etc., for depths of if, 2, 2^, 2^, 2j, 3, etc., fathoms respectively.
If the line is graduated in feet leather tags are used every
five feet, the intermediate foot-marks being cotton or woollen
strips. The ten-foot tags are notched with one, two, three,
etc., notches for the 10, 20-, 30-, etc., foot points, up to fifty
feet. The fifty-foot tag may have a hole in it, and the 60-, 70,
80, etc., foot-marks have tags all with one hole and with one,
two, three, etc., notches. The intermediate five-foot points
have a simple leather tag unmarked. Sometimes the figures
are branded on the leather tags, but notches are more easily
read. The zero of the graduation is the bottom of the lead.
The leather tags are fastened into the strands of the line ; the
cloth strips may be tied on. The line should be frequently
tested, and if it changes materially a table of corrections
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HYDR0GRAPH2C SURVEYING, 3^3
should be made out and all soundings corrected for erroneous
length of line.
236. Sounding-poles should be used when the depth is
less than about fifteen feet. The pole may be graduated to
feet simply, or to feet and tenths, according to the accuracy
required.
237. Making Soundings in Running Water. — The
sounding-boat should be of the " cutter" pattern, with a sort
of platform in the bow for the leadsman to stand on. If the
current is swift, six oarsmen will be required and two ob-
servers and one recorder. One of the observers may act as
steersman. If the depth is not more than sixty or eighty feet,
the soundings are made without checking the boat, the leads-
man casting the lead far enough forward to enable it to reach
bottom by the time the line comes vertical. When the depth
and the current are such as to make this impossible, the boat
is allowed to drift down with the current and soundings taken
at intervals without drawing up the lead. The boat is then
pulled back up stream and dropped down again on another
Hne, and so on.
In still water a smaller crew and outfit may be used, as the
boat may be stopped for each sounding if necessary.
The record should give the date, names of observers, general
locality, number or other designation of line sounded, the
time, the two angles, the stations sighted, and the depth for
each sounding, and the errors of the graduated lengths on the
sounding-line.
238. The Water-surface Plane of Reference. — In order
to refer the bottom elevations to the general datum-plane of
the survey, it is necessary to know the elevation of the water-
surface at all times when soundings are taken. In tidal waters
the elevation of " mean tide*' is the plane of reference for both
the topographical and hydrographical surveys, and then the
state of the tide must be known with reference to mean tide.
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3^4 SURVEYING,
This is found from the hourly readings of a tide-gauge (pro-
vided it is not automatic), the elevation of the zero of which,
with reference to mean tide-water, has been determined. All
soundings must then be reduced to what they would have been
if made at mean tide before they are plotted.
If the soundings are made in lakes, the datum is usually
the lowest water-stage on record ; and here also gauge-readings
are necessary, as the stage of the water in the lake varies from
year to year. In this case the gauge need only be read twice
a day.
In rivers of variable stage the datum is either referred to
mean or low-water stage, or else to the general datum of the
map. If the stage is changing rapidly the gauge should be
read hourly when soundings are taken, otherwise daily readings
are sufficient. If the soundings are to be referred to the
general datum of the map, then the slope of the stream must
be taken into account. If they are referred to a particular
stage of water in the river, then the slope does not enter as a
correction, as the slope is assumed to be the same at all stages,
although this is not strictly true.
239. Lines of Equal Depth correspond to contour lines
in topographical surveys; but to draw lines of equal depth
with certainty the elevations of many more points are neces-
sary than are needed for drawing contour lines, because the
bottom cannot be viewed directly, while the ground can be.
Where the ground is seen to be nearly level no elevations
need be taken, while for a similar region of bottom a great
many soundings would be required to prove that it was not
irregular.
240. Soundings on Fixed Cross-sections in Rivers.—
Where the same section is to be sounded a great many times,
and especially when it is desirable to obtain the successive
soundings at about the same points, it is best to fix range
posts on the line of the section (on both sides if it be a river)
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IIYDROGRAPHIC SURVEYIXG.^
305
and then fix one or more series of intersecting ranges at points
some distance above or below the section on one or both sides
of the river. The soundings can then be made at the same
points continuously without having to observe any angles at
all. Such a system of ranges is shown in Fig. 72. A A' and
BB' are range-poles on the section line. O and O' are tall
white posts set at convenient points on opposite sides of the
river, either above or below the section. I., II., III., etc., are
shorter posts set near the bank in such positions that the in-
tersection of the lines 0-1 , (7-1 1., etc., with the section range
Fig. 72.
BB will locate the soundings at i, 2, etc., on this section line.
The posts in the banks should be marked by strips nailed upon
them so as to make the Roman numerals as given in the figure.
Such a system of ranges as the above is useful also for fixing
points on a section-line, for setting out floats, or for running
current meters for the determination of river discharge,
241. Soundings for the Study of Sand-waves. — In all
cases where streams flow in sandy beds, the bottom consists
of a series of wave-like elevations extending across the chan-
nel. These are very gently sloping on the up-stream side
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and quite abrupt on the lower side. They are called sand-
waves, or sand-reefs. They are constantly moving down-
stream from the slow removals from the upper side and accre-
tions on the abrupt lower face. They have been observed as
high as ten feet on the Mississippi River, and with a rate of
motion as great as thirty feet per day. In order to study the
size and motion of these sand-waves, it is necessary to take
soundings very near together, on longitudinal lines over the
same paths at frequent intervals for a considerable period.
The boat is allowed to drift with the current and the lead floats
with the boat near the bottom. It is lowered to the bottom
every few seconds and the depth and time recorded. About
once a minute the boat is located by two instruments on shore
or in the boat, and so the exact path of the boat located. A
profile of the bottom can then be drawn for the path of the
boat. A few days later the same line is sounded again in a
similar manner and the two profiles compared. It will be
found that the waves have all moved down-stream a short dis-
tance, the principal waves still retaining their main charac-
teristics, so that identification is certain.*
242. Areas of Cross-section are obtained by plotting
the soundings on cross-section paper, the horizontal scale be-
ing about one tenth or one twentieth of the vertical. The
horizontal line representing the water-surface is drawn, and the
plotted soundings joined by a free-hand line. The enclosed
area is then measured by the planimeter. If the horizontal
scale is 50 feet to the inch and the vertical scale 5 feet to the
rinch, then each square inch of the figure represents 250 square
feet of area. The planimetci should be set to read the area
in square inches, and the result multiplied by 25o.f
* It is believed the author made the first successful study of the size and
rate of motion of sand-waves, at Helena, Ark., on the Mississippi River, in
1879. See Rep. Chief of Engineers, U. S. A., 1879, vol. iii., p. 1963.
f See p. 143 for a description and theory of the planimeter.
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HYDROGRAPHIC SURVEYING, 307
Areas of cross-section are usually taken in running water,
and here great care must be taken to get vertical soundings,
and to make the proper sounding-line corrections. They
should be taken near enough together to enable the bottom
line to be drawn with sufficient accuracy.^
BENCH-MARKS, GAUGES, WATER LEVELS, AND RIVER-SLOPE.
243. Bench-marks should be set in the immediate vi-
cinity of each water-gauge, and these connected by duplicate
lines of levels with the reference-plane of the survey. If the
gauge is not very firmly set, or if it is necessary to move it fo.
a changing stage, its zero must be referred again to its bench-
mark by duplicate levels, whenever there is reason to suspect
it may have been disturbed. Such bench-marks as these are
usually spikes in the roots of trees or stumps.
244. Water-gauges are of various designs, according to
the situation and the purpose in view. For temporary use
during the period of a survey, a staff gauge is best, consisting
of a board painted white, of sufficient length, graduated to feet
and tenths in black. Sometimes it is graduated to half-tenths,
but this is useless unless in still water, and there is never any
need of graduation finer than this. The gauge maybe read to
hundredths of a foot if the water is calm enough. It should
be nailed to a pile or to a stake driven firmly near the water's
edge. It is read twice a day, or oftener, if the needs of the
service require.
For the continuous record of tidal stages an automatic, or
self-registering, gauge is employed. For rivers with widely
varying stage an inclined scantling is fixed to stakes set from
low to high water along up the sloping bank. It should be
placed at a point where the bank is neither caving away nor
growing by filling-in of new deposits. After the scantling is
set (the slopes not necessarily the same throughout its length),
the foot and tenth graduations are set by means of a level and
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308 SURVEYING,
marked by driving copper tacks. The automatic gauge is
described in Chap. XIV. The staff gauge is the one generally
used for engineering and surveying purposes.
245. Water-levels. — The surface of still water is by defi-
nition a level surface. This fact is used to great advantage
on the sea-coast, on lakes, ponds, and even on streams of little
slope or on such as have a known slope. Thus, in finding the
elevations of the Great Lakes above the sea-level, the elevation
above mean tide-water of the zero of a certain water-gauge at
Oswego, N. Y., on Lake Ontario, was determined. Then the rela.
tive elevations of the zeros of certain gauges at Ports Dalhousie
and Colborne, at the lower and upper ends of the Welland
Canal respectively, were found by levelling between them, thus
connecting Lake Ontario with Lake Erie. Lakes Erie and Hu-
ron were joined in a similar manner by connecting a gauge at
Rockwood, at the mouth of the Detroit River with one at Lake-
port, at the lower end of Lake Huron. Lakes Michigan and
Huron were assumed to be of the same level on account of
the small flow between them and the very large sectional area
of the Straits of Mackinac. Finally, a gauge at Escanaba, on
Lake Michigan, was joined by a line of levels with one at Mar-
quette, on Lake Superior. This completed the line of levels
from New York to Lake Superior, when sufficient gauge-read-
ings had been obtained to enable water lei^elsio be carried from
Oswego to Port Dalhousie, on Lake Ontario ; from Port Col-
borne to Rockwood, on Lake Erie • and from Lakeport, on
Lake Huron, to Escanaba, on Lake Michigan. It was found
that these water-levels were very accurate. Relative gauge-
readings were compared for calm days, as well as for days
when the wind was in various directions, and a final mean
value found which in no case had a probable error as great as
0.1 foot.*
• See Primarv Triangnlatioc of the U. S. Lake Survey.
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HYDROGRAPHIC SURVEYING, 3O9
A line of levels run along a lake shore or canal in calm
weather should be checked at intervals by reading to the
water-surface, and in a topographical survey the stadia-rod
should frequently be held at the water-surface, even when the
body of water is a stream with considerable slope, as it gives.
a check against large errors even then, and at the same time
gives the slope of the stream. Mean sea-level at all points
on the sea-coast is universally assumed to define one and the
same level surface. It is probable, however, that this is not
strictly true. Wherever a constant ocean current sets stead-
ily against a certain coast, it would seem that the water here
must be raised by an amount equal to the head necessary to
generate the given lost motion. If the current flows into an
enclosed space, as the equatorial current into the Gulf of
Mexico, or the tides into the Bay of Fundy, the water-surface
may rise much higher. There is some evidence that the ele-
vation of mean tide in the Gulf of Mexico is two or three inches
higher than that of the Atlantic at Sandy Hook. The evi-
dence on this point is as yet insufficient to warrant any certain
conclusion, however.
246. River Slope is a very important part of a river survey.
Sometimes it is desirable to determine it for a given stretch
of river with great care, in which case it is well to set gauges
at the points between which the slope is to be found and con-
nect them by duplicate lines of accurate levelling. The gauges
are then read simultaneously every five minutes for several
hours and the comparison made between their mean readings.
This is always done in connection with the measurement of the
discharge of streams when the object is to find what function
the discharge is of the slope. It is now known, however^ that
in natural channels the discharge is no assignable function of
the slope, as iz e:L;;lainci in section 259. For ordinary purposes
the river slope may be determined with sufficient accuracy by
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3IO SURVEYING.
simply reading the level or the stadia-rod at water-surface as
the survey proceeds, daily readings of stage being made at
permanent gauges at intervals of fifty miles or less along the
river.
In all natural channels the local slope is a very variable
quantity. It is frequently negative for short distances in cer-
tain stages, and over the same short stretch of river it may
vary enormously at different stages, and even for the same
stage at different times. It is determined by the local channel
conditions, and these are constantly changing in streams flow-
ing in friable beds and subject to material changes of stage.
Great caution must therefore be exercised in introducing it
into any hydraulic formulae for natural channels. It is usually
expressed as a fraction, being really the natural sine of the
angle of the surface to the horizon. That is, if the slope is one
foot to the mile it is yg^g^ = 0.000189.
THE DISCHARGE OF STREAMS.
247. Measuring Mean Velocities of Water Currents.
This is usually done only for the purpose of obtaining the
discharge of the stream or channel, but sometimes it is done
for other purposes, as for the location of bridge piers or harbor
improvements. In the case of bridge piers the direction of
the current at dififerent stages must be known, so that the piers
maybe set parallel to the direction of the current. For find-
ing the discharge of the stream or other channel the object may
be:
(i) To obtain an approximate value of the discharge at the
given time and place.
(2) To obtain an exact value of the discharge at the given
time and place.
(3) To obtain a general formula from which to obtain sub-
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HYDROGRAPinC SURVEYING, 3II
sequent discharges at the given place, or to test the truth of
existing formulae, or to determine the relative efficiency of
certain appliances or methods.
It will be assumed that the second object is the one sought,
and modified forms of the methods used to accomplish this
may be chosen for other cases.
The mean velocity of a stream is by definition the total dis-
charge in cubic feet per second divided by the area of the
cross-section in square feet. This gives the mean velocity in
feet per second. Evidently this is the mean of the veloci-
ties of all the small filaments (as of one square inch in area) on
the entire cross-section. If the velocities of these filaments
could be simultaneously and separately observed and their
mean taken, this would be the mean velocity of the stream. It
is quite impossible to do this ; but the nearer this is approached,
the more accurate is the final result. If, however, we could
obtain by a single observation the mean velocity of all the fila-
ments in a vertical plane, the number of necessary observations
would be diminished without diminishing the accuracy of the
result. There are two common methods of measuring the ve-
locities of filaments at any part of the cross-section, and one
for obtaining at once the mean velocity in a vertical plane.
These are by sub-surface floats and current-meters, and by rod
floats, respectively.
248. By Sub-surface Floats.— The ideal sub-surface float
consists of a large intercepting area maintained at any depth
in a vertical position by means of a fine cord joined to a sur-
face float of minimum immersion and resistance, which bears
a signal-flag. As good a form as any, perhaps, for the lower
float, or intercepting plane, consists of two sheets of galvanized
iron set at right angles, and intersecting in their centre lines, as
shown in Fig. 73. There are cylindrical air-cavities along the
upper edges and lead weights attached to the lower edges of
the vanes. These serve to give the desired tension on the
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312
S UK V EYING.
connecting cord and to maintain the float in an upright posi-
tion, even though the cord is drawn out of the vertical by
faster upper currents. The vanes should be from six to fifteen
inches in breadth by from eight to twenty inches high, accord-
ing to the size of the stream. The circular ribs serve simply
to hold the vanes in place. The upper float is hollow, cylin-
Fic. 73.
Jrical in plan, and carries a small flag. The tension on the
cord should be from one to five pounds, according to the size
of the floats. The cord itself should be of woven silk and as
small as possible, so as to exercise a minimum influence on the
motion of the lower float. Wire is not suitable for this pur-
pose, as it kinks badly in handling. The theory is that the
lower float will move with the water which surrounds it, and
that the upper float will be accelerated or retarded according
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HYDROGRAPHIC SURVEYING. 313
as the surface current is slower or faster than that at the sub-
merged float. The velocity of the current at any depth can
thus be determined by running the lower float at this depth
and observing the time required for the upper float to pass
between two fixed range-lines at right angles to the direction
of the current about two hundred feet apart. The floats are
started about one hundred feet above the upper range-line, and
picked up after having passed the lower range. Two transits
are usually used for locating and timing the floats, one being
set on each range. When the float approaches the upper
range the observer on this line sets his telescope on range and
calls ** ready" as the float enters his field of view. The other
observer then clamps his instrument and follows the float with
the aid of the slow-motion or tangent screw. When the float
crosses the vertical wire of the upper instrument he calls ** tick,"
and the lower observer reads his horizontal angle. He then
sets his telescope on the lower range while the upper observer
follows the float with his telescope, and the operation is re-
peated to obtain an intersection on the lower range. One or
two timekeepers are needed to note the time of the two
"tick" calls, the difference being the time occupied by the
float in passing from the upper to the lower range-line. Both
these signals are sometimes transmitted telegraphically to a
single timekeeper. When the angles are plotted the path of
the float is also obtained.
If the channel is not too wide, wires may be stretched
across the stream and the float stations marked on these, or
the float stations may be determined by means of fixed ranges
on shore. The passage of the floats across the section lines
may then be noted by a single individual without a transit,
using a stop-watch and possibly a field-glass. He starts the
watch when the float reaches the upper section, walks to the
lower section, and stops the watch when the float passes this
range-line. The near range consists of a plumb-line, or wire
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SURVEYING,
suspended vertically; and the observer stands several feet back
of this, and brings it in line with the range-post on the opposite
side of the stream.
If several floats are started a few minutes apart at the same
station and at the same depth, they will sometimes vary as
Fig. 74.
Oiuch as twenty per cent in their times of passage, showing
great irregularity in the velocity of different parts of the same
filament. This is due to internal movements in the water,
such as *' boils,*' eddies, etc. It is for this reason that great
refinement in such observations is useless. A float observation
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HYDROGRAPHJC SURVEYING,
31.^
gives only the velocity of a given small volume of water which
surrounds the lower float, while a current-meter observation,
as will be seen, gives the mean velocity of a given filament of
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3l6 SURVEYING,
the stream of any required length. And as different portions
of the same filament have very different longitudinal velocities,
it requires a great many float observations to give as valuable
information as may be obtained by running a current meter
in the same filament for one minute.
If discharge observations are to be repeated many times at
the same sections, then an auxiliary range should be established
from which to start the floats; and if it is desirable to always
run them over the same paths, these may be fixed by means
of a system of intersecting ranges as described on p. 305.
249. By Current-meter.* — ^This is the most accurate method
of obtaining sub-surface velocities ever yet devised. Three
patterns of current-meters are shown in Figs. 74 and 75.
The first and third are shown in elevation, together with the
electrical recording-apparatus. The second is shown in plan.
The first has helicoidal and the other two conical cup-shaped
vanes. Neither has any gearing under water, the record being
kept by means of an electrical circuit which is made and bro-
ken one or more times each revolution. The cup vanes are
better adapted to water carrying fibrous materials which tend
to collect on the moving parts. The friction can also be made
less on the cup meters, agate or iridium bearings being used.
The recording-apparatus is kept on shore or in a boat, while
the meter is suspended by proper appliances at any point of
the section at which the velocity of the current is to be measured.
In deep water a boat, or catamaran, is anchored at the desired
♦ Invented by Gen. Theo. G. Ellis, and first used on the survey of the
Connecticut River. The telegraphic attachment is due to D. Farrand Henry
of Detroit, Mich. See Report of the Chief of Engineers, U. S. A., 1878, p. 30S.
The form shown in Fig. 75 is due to W. G. Price, and was specially de-
signed to be used on the Mississippi River. It is very strong and well pro-
tected against floating drift. The first two forms are manufactured by Buflf
& Berger, of Boston, while the Price meter is made by W. & L. E. Gurley,
Troy. See also the Ritchie & Haskell meter for direction aud velocity of sub*
currents. Art. 255.
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HYDROGRAPHTC SURVEYING. 37/
point, and a weight attached to the meter, which is then lowered
to the requisite depth by means of a windlass. After it is in
place the connection is made with the battery, and the record
kept for a given period of time, as for two or three minutes.
If the operation is to be repeated often at the same section a
wire anchorage laid across the stream above the line would be
found useful. This wire is anchored at intervals and is used
both for holding the boat (or catamaran) in place and for pull-
ing it back and forth across the stream. In large rivers a
steam-launch may be required for handling the catamaran.*
In this case the record begins and ends when the observer is
brought on range, it being impossible to hold up steadily
against the current. If only the discharge of the stream is
sought, the meter is run at mid-depth at a sufficient number
of points in the section.
The mean velocity in a vertical section at a given point may
be obtained by moving the meter at a uniform rate from sur-
face to bottom and back again, noting the reading of the regis-
ter for the two surface positions, and also for the bottom posi-
tion. If the boat was stationary and.the rates of lowering and
raising strictly constant and equal, the number of revolutions in
descending and in ascending should be equal. Either of these
registrations, divided by the time, would give the mean regis-
tration per second of all the filaments in that vertical plane.
The mean of the downward and upward results may be used
as giving the mean velocity in that vertical plane. This will
not be quite accurate, since it is impossible to run the meter
very close to the bottom, but the results will be found useful
for comparison with the mid-depth results. Such observations
are sometimes called integrations in a vertical plane.
250. Rating the Meter. — When any kind of current-meter
♦For a description of the latest methods used in gauging the Miss:«»«iippi
River see Report of the Miss. Riv. Cpm. fj^iSSs, Appendix F.
21
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SURl^EVlMO.
is used for determining the velocity of passing fluids, only the
number of revolutions of the wheel carrying the vanes is ob-
served for a given time. Before the velocity of the fluid in feet
per second can be found, the relation between the rate of revo-
lution of the wheel and the rate of motion of the fluid must be
determined for all velocities that are to be observed. The de-
termination of this relation is called rating the meter. It is usu-
ally done by causing the meter to move through still water at
Fic. 76.
a uniform speed, and noting the time occupied and the corre-
sponding number of registrations made in passing over a given
distance. It may be attached to the prow of a boat, as shown
in Fig. 76, the electric register being in the boat. The dis-
tance divided by the time gives the rate of motion or velocity
of the meter through the water. The number of registrations
(revolutions of the wheel) divided by the time gives the rate
of motion of the wheel. The ratio of these two rates is the
: coefficient by which the registrations of the meter are trans-
formed into the velocity of the current. This ratio is not a
constant, but is usually a linear function of the velocity. Thus,
if the observations be plotted, taking the number of registra-
tions per second as abscissae and the velocities in feet per
second as ordi nates, they will be found to fall nearly in a right
line, the equation of which is
y z=, ax -\-b
(0
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HYDROGRAPHIC SURVEYING. 319
Here x and y are the observed quantities, while a and b are
constants for the given instrument. If these constants could
be found, then the values of y (velocity) could be obtained for
all observed values of x (registrations). There are two ways of
solving this problem — one graphical and one analytical. Evi-
dently any two observations at different speeds would give
values of a and b\ but to find the best or most probable values
of these constants a great many observations are taken, so that
we have many more observations than we have unknown quan-
tities. Each pair of observations would give a different set of
values of a and b. The most convenient method of finding
the most probable values of these functions, though somewhat
approximate, is
(i) The Graphical Method of Solution. — This consists simply
in plotting the corresponding values of x and y on coordi-
nate paper, and drawing the most probable straight line through
the points. Then the tangent of the angle this line makes
with the axis of x is a, and the intercept on the axis of y is b.
One point on this most probable line is the point {x^y^, x^ and
y^ being the mean values of the coordinates of all the plotted
points. This is shown by equation (3). Having determined
this point, a thread may be stretched through it and swung
until it seems to be in a position of equilibrium, when each
point is conceived as an attractive force acting on the line, the
measure of the force being the vertical intercept between the
point and the line. The arms of these forces are evidently
their several abscissae. Or the forces may be measured by
their horizontal intercepts, and then their arms are their seve^
ral ordinates. For the position of equilibrium the sum of the
moments of these forces about the point {x^y^ would be zero.*
Such a determination of a and b would be found sufficiently
accurate for all practical purposes, but if desired the problem
may be solved by
* All this simply means to fix this most probable line by eye, through the
point (jfo^o), giving greatest weight to the extreme points.
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320 SU/^P'EYIA'G,
(2) The Rigid or Analytical Method. — Equation (i) may be
written
b'\-xa- y = o.
Every observation may be written in this form, these being
called the observation equations. It is probable that no given
values of a and b would satisfy more than two of these obser-
vations ; and if the most probable values be used, there would,
in general, be no single equation exactly satisfied. If we let
x^y ;r„ etc., ^j, j„ etc., and z/„ t;„ etc., be the several values of
Xyjf, and the corresponding residuals for the several observa-
tion equations, we would have
b + x^a—y^z=v^.
(2)
Since b enters alike in all of them, it is evident that these
equations are all of equal value for determining b. Also, since
the properly weighted arithmetic mean is the most probable
value of a numerously observed quantity, and since in this case
the equations (or observations) have equal weight for deter-
mining b, we may form from the given series of equations a
single standard or " normal ** equation which will be the arith-
metic mean of the observation equations : put this equal to zero
and say this shall give the value of b. If x^ and y^ be the mean
values of the observed x*s and ys, we would then have, by add-
ing the equations all together and dividing by their number
b + x,a^y,=iO,orb=y.-x^ ... (3)
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HYDROGRAPHIC SURVEVIA'G, ^21
Substituting this value of b in equation (2), we have
.... (4)
{x, - x,)a - {y, -y,) = v, ;
{^m - ^0)^ - {ym-y,) = Vn,
We here have a series of equations involving one unknown
quantity ; but they evidently are not of equal value in deter,
mining the unknown quantity /?, since its coefficients are very
different. In fact, the relative value of these equations for de-
termining a is in direct proportion to the size of this coefficient,
so that if this coefficient is twice as large in one equation as in
another, the former equation has twice the value of the latter
for determining a. In other words, they should all be weighted
in proportion to the values of these coefficients, and a conve-
nient way of doing this is to multiply each equation through
entire by this coefficient. The resulting multiplied equations
then have equal weight, and may then be added together to
produce another ** normal ** equation for finding a. This result-
ing equation is
l{x-x,y]a-[{x-x, (/-:»'.)] =0. ... (5)
where [ ] is a sign of summation. If we had divided this
equation by the number of observation equations m, it would
in no sense have changed it so far as the value of a is concerned.
From equation (5) we can find the mean or most probable
value of a, which when substituted in (3) gives the most prob-
able value of b. These values should agree very closely with
those found by the graphical method. The analytical method
here given is precisely that by least squares, though arrived at
through the conception of a properly weighted arithmetic mean,
instead of by making the sum of the squares of the residuals a
minimum.
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322
SURVEYING.
The following is an actual exannple from the records of the
Mississippi River Survey:
REDUCTION OF OBSERVATIONS FOR RATING METER A.
taken at Paducah, Ky., June 21, 1882.
W. G. Price, Obsei-ver, L. L. Wheeler, Computer.^
No.
r
t
X
y
x-^o
y-yo (-^-.ro)«
Ky-y^)
Remarks.
I 100
53
I 886
Z'llA
+ 0.117
+ 0.245
+ 0.014
+ 0.029
Observations
aioi
44
2.293
4.544+ 0.526
+ I. 015
+ 0.277
+ 0.534
made with
3101! 41
2.464
4.878
+ 0.695
+ 1.349
+ 0.483
+ 0.938
meter on vertical
4 96124
0 774
1. 613
- 0.995
— I. 916
+ 0.990
+ 1.906
iron rod, five
5 94152
0.618
1. 316
- I. 151
— 2.213
+ 1.325
+ 2.548
feet in front of
6 go IQ3
0.466
1.036
- 1.303
- 2.493
+ 1.697
+ 3.249
bow of skiff, m
7 9'
181
0.503
1. 105
- 1.266
- 2.424
+ 1.603
+ 3.0691 pond.
8 103
28
3.678
7.142
+ 1.909
+ 3.613
+ 3.644
+ 6.903
9 100
53
1.886
3 774
+ 0.117
+ 0.245
+ 0.014
+ 0.029
LePjft-i A bir».
10 98 73
1.342
2.740
- 0.427
- 0.789
4- 0.182
+ 0.337
= 200 fCtft.
II 103
29
3-552
6.896
+ 1.783
+ 3-3^1
+ 3.178
+ 6.002
[^
] =
19.464I38.818 -\y\
[{x-'x.n
= 13.407
25.544=[(.^-JroK/-.y'.;)
X
0 ■=
1.769
3.529
-y^
Normal Equations.
b + 1.769a — 3529 = o ; Whence a = 1.905 ;
13.407^ - 25.544 = o. ^ = 0.159.
Equation for Rating,
y = 1.905^^ + 0.159.
Even where the analytical method is to be used it is al-
ways well to plot the observations for purposes of study.
Then if any observations are especially discrepant, the fact will
appear. By consulting column six of the computation it will
♦ In tlie original computation the method by least squares was used and thtt
probable errors of a and ^ iound.
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HYDkOGRAPMlC SURVEYING, 323
be seen that observations of greatest weight were those taken at
very high and at very low velocities. If the observations were
taken in three groups about equally spaced, an equal number
of observations in each group, the members of a group being
near together, then the mean of each group could be used as
a single observation. The middle group would serve to show
whether or not the unknown quantities were linear functions of
each other, since, if they were, the three mean observations
should plot in a straight line. The value of a could be com-
puted from the two extreme mean observations, and the value
of b from the mean of all the observations as before. This
would give a result quite as accurate as to treat them separately.
If the observations do not plot in a straight line, draw
the most probable line through them, and prepare a table of
corresponding values of x and y from this curve. In any case,
a reduction table should be used.
The meter should be rated frequently if accurate results are
required. In the rating the meter should be fastened several
feet in front of the bow of the boat, and in its use it should be
run at a sufficient distance from the boat or catamaran to be
free from any disturbing influence on the current.
251. By Rod Floats. — These may be either wooden or tin
rods, of uniform size, loaded at the- bottom, and arranged for
splicing if they are to be used in deep water. If the channel
were of uniform depth, and the rod reached to the bottom with-
out actually touching, then the velocity of the rod would be
the mean velocity of all the filaments in that vertical plane,*
* This is not strictly true, since the pressure of a fluid upon a body moving
through it varies as the square o! its relative velocity. The rod moves faster
than the bottom filaments and slower than the upper filaments, but this differ-
ence is greatest at the bottom. Therefore, the retarding action of the bottom
filaments will have undue weight, as it were, and so the velocity of the rod will
really be about one per cent slower than the mean velocity of the current See
*• Lowell Hydraulic Experiments," by James B. FrancU.
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324 SURVEYING,
and this is the value sought. In practice the rod can never
reach the bottom, even in smooth, artificial channels, while in
natural channels the irregularities are usually such as prohibit
its use within several feet of the bottom. The methods of
observation are the same as with the double floats, and their
velocity is the mean velocity of the water in that plane to the
depth of immersion. For artificial channels, and for natural
channels not more than twenty or thirty feet deep, rod floats
may be advantageously used. Beyond that depth they cannot
be made of sufficient length to give reliable results. The
method is, therefore, best adapted to artificial channels of uni-
form cross-section.
The immersion of the rod should be at least nine tenths of
the depth of the water, in which case, and for uniform channels,
as wooden flumes, Francis found that the velocity of the rod
required the following correction to give the mean velocity of
the water in that vertical plane :
Vm= Vr [i-o.ii6(VZ>-o.i)].
Where Vm = mean velocity in vertical plane;
Vr = observed velocity of rod ;
n _ depth of water below bottom of rod
depth of water
For natural channels, or for a less immersion than nine-
tenths of the depth the formula cannot be used with certainty.
The rods should be put into the water at least twenty feet
above the upper section.
252. Comparison of Methods.— (i) The method by double
floats is adapted to large and deep rivers, or rapid currents
carrying much drift or impeded by traffic. It may be used
in all cases, but it has the disadvantage of registering only the
velocity of a small volume of water surrounding the lowei
float.
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HYDROGRAPHIC SURVEYING. 3^5
(2) The method by meters is adapted to large or small
streams. It records the mean velocity of a filament of indefinite
length : but it cannot be used where the water carries consider-
able floating debris, or where the current is too swift to admit
of a safe anchorage.
(3) The method by rods is best adapted to small channels
of uniform section ; it records the mean velocity in a vertical
plane to a depth equal to its immersion, and it can be univer-
sally used when the law of the velocities in a vertical plane is
known, for then a proper coefficient could be derived for any
depth of immersion.
(4) One rod observation of sufficient immersion is prob-
ably as good as several float observations, and a current-meter
observation of two or three minutes is worth as much as
twenty float observations for the same filament, provided the
meter's rate is constant and well determined.
(5) The rods and floats are cheaper in first cost than the
meter ; but if the work is to be prosecuted for a considerable
period, the excess in the cost of the outfit will be more than
balanced by the diminished cost of the work, by using the
meter. On the whole, it may be said that the method by cur-
rent-meter is the most accurate and satisfactory of any yet de-
vised for measuring the velocity of running water.
253. The Relative Rates of Flow in Different Parts of
the Cross-section.— (i) In a horizontal plane. If the cross-
section of a stream were approximately the segment of a circle,
then the relative rates of flow of the different filaments in any
horizontal plane would be very nearly represented by the ordi-
nates to a parabola, the axis of the parabola coinciding with
the middle of the stream. If there should be any shoaling in
any part of this ideal section the corresponding ordinates would
be shortened, so that when the curve of the bottom is given
the curve of velocities in a horizontal plane can be fairly pre-
dicted. This applies only to straight reaches. 11 a portion of
the section has a flat bottom line, the velocities over this por-
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326
SURVEYING,
tion will be about uniform. Where the depth is changing rap-
idly on the section, there the velocities will be found to chang<?
rapidly for given changes in positions across the section.
It follows from this that observation stations should be
placed near together where the section has a sloping bottom
line, and they may be placed farther apart where the bottom
line of the section is nearly flat. They are usually put closer
together near the bank than near the middle ot the stream.
(2) In a vertical plane. A great deal of time and talent has
been spent in trying to find the law of the relative rates of flow
0 2 8 4
5 6 7 8
r^Ti^i^oiTr^
: I
I_
- J
f
^lOlVfii'l t (> —
' ■ 7 -
i
h
^7
- 71
t.
i^'-i,
U4
C r 2' 8' 4
ILL .
Fig. 77.
in a vertical plane, but there is probably no law of universal
application. The curve representing such rates of flow will
always resemble a parabola more or less, the axis of which is
always beneath the surface except when the wind is down
stream at a rate equal to, or greater than, the rate of the cur-
rent. That is to say, the maximum velocity is always below
the surface except where the surface filaments are accelerated
by a down-stream wind, and it is generally found at about one
third the depth. The cause of this depression of the filament
( f maximum velocity is partly due to the friction of the air.
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HYDROGRAPHJC SURVEYING.
327
but mostly to an inward surface flow from the sides toward
the centre, which brings particles having a slower motion
towards the middle of the surface of the stream. This inward
surface flow is probably due to an upward flow at the sides
caused by the irregularities of the bank, which force the parti-
cles of water impinging upon them in the direction of the least
resistance which is vertical.* The curves in Fig. ^^ represent
the mean vertical curves of velocity observed at Columbus,
Ky., on the Mississippi River and given in Humphreys and
Seal€ 4 reef. ^
9_ J f I T
Fig. 78.
Abbot's Report. The left-hand vertical line is the axis of ref-
erence, and the curves are found to fall between the seven- and
eight-foot lines. That is, the velocity at all depths in this
plane was between seven and eight feet per second. In this
case double floats were used, and it is probable that the bottom
velocities were not very accurately obtained. The eflect of the
wind is here shown in shifting the axis of the curve. It is to
* Sec paper by F. P. Stearns before the Am. Soc. Civ. Engrs., vol. xii. p. 331.
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328 SURVEYING.
be observed that these curves all intersect at about mid-depth.
That is to say, the velocity of the mid-depth filament is not
affected by wind. This is why the mid-depth velocity should
be chosen when the velocity of but a single filament is to be
measured, and from this the mean velocity in the vertical sec-
tion derived. It has also been found that the mid-depth veloc-
ity is very near the mean velocity, being from one to six per
cent greater, according to depth and smoothness of channel.
In general, for channels whose widths are large as compared
to their depths, a coefficient of from .96 to .98 will reduce
mid-depth velocity to the mean velocity in that vertical plane.
In Fig. 78* are shown the relative velocities in different parts
of the Sudbury River Conduit of Boston. The velocity at
each dot was actually measured by the current-meter. The
lines drawn are lines of equal velocity, being analogous to con-
tour lines on a surface, the vertical ordinates to which would
represent velocities. The method of obtaining these velocities
is shown in Fig. 79. B is a pivoted sleeve through which the
meter-rod slides freely. At A there is a roller fixed to the rod
which runs on the curved tracks a a a. The graduations on
these tracks fix the different positions of the meter, these be-
ing so spaced that they control equal areas of the cross-section.
Integrations were here taken in horizontal planes by moving the
meter at a uniform rate horizontally.
254. To find the Mean Velocity on the Cross-section.
— It is evident that this mean velocity cannot be directly ob-
served. In fact, it can only be found by first finding the dis-
charge per second and then dividing this by the total area of
the section. That is to say, the mean velocity is, by definition,
-I
* This and the following figure are taken from the paper by F. P. Steams,
mentioned in foot-note on the previous page.
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HYDROGRAPHIC SURVEYING.
329
r
?v;f\^
/%3
W^
eS^-r
I ' ^ »= ^ ^
1
5:
* *
1'
•
-
in
»?
■
#
y^
r_
'^
#
-
'
1
LpJ^
.J.
* •
E
! :
'^1 -
'
■
-
ifi
mr
V
•
•
*V
J
•
-
. 'i*
'
" ||-
-/
■
-
"- '
H'
■
1 ,
*
Ti
* ;
* 1
* p
-
*
' f '*'
-
'
-
■
r;i
'
V
*
-
^»
.
•
-» -
■
'
T
'.'
- 1
'
' '■
V
•
■1-
•
"
3
* 1 '
V
-
■
■|(
-
V ,
•
'
. V
L^
■
L^_
■
*
I'-J
■1
-
■r
1
'
«
•
, «
V
■
]■
-
-
1
.*
i'
VjL
■
■i
m
r
1
•'!■
•
1
-
*/
*
^^
^
■w
u
'i
1-
L-
i
f
t'
1,
m
iv
3
Fig. 79.
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330 SURVEYING,
The area of the section is found by means of properly located
soundings. The actual velocities of certain filaments crossing
this section are then observed, and the section subdivided in
such a way that the observed velocities will fairly represent
the mean velocities of all the similar filaments (usually mid-
depth) in that subsection. Each observed velocity is then
reduced to the mean velocity in that vertical plane, and this is
assumed to be the mean velocity in that subsection. These
mean velocities, multiplied by the areas of their corresponding
sections, give the discharges across these sections, and the sum
of these partial discharges is the total discharge, (2> ^^ the
above equation. This may be shown algebraically as follows :
Let Fj, F„ F„ etc., be the observed velocities ;
C the coefficient to reduce these to the mean velocity
in a vertical plane ;
^„ A^, A^, etc., the partial areas of the cross-section
corresponding to the observed velocities F,, V,,
F., etc.;
A the total area of the cross-section = A^'\- A^'\- A^
etc.;
(2i» Qi> (2.1 etc., the partial discharges;
Q the total discharge ;
V the mean velocity for the entire section.
Then Q, = CV,A, ; Q, = CV,A,. etc. ;
Q=Q. + Q.+ etc. ^ C{Ay, 4- A,V, + etc.);
and ^ = f - 2^Ay, + A^V, + etc.).
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HYDROGRAPHIC SURVEYING. 33 1
It has been here assumed that observations are made at but
one point in any vertical plane. The method is the same, how-
ever, in any case, it only being necessary to apply such a co-
efficient to the observed velocity as will reduce it to the mean
velocity in its own sub-area. If these partial areas are made
small, as in the case of the Boston Conduit, the observed ve*
locities may be taken as the mean velocities in those areas ; and
if these areas are all equal, which was also the case in this con-
duit, then the mean velocity is the arithmetic mean of all the
observed velocities. The partial and total areas are best found
by means of the planimeter, the cross-section having been
carefully plotted on coordinate paper.
255. Sub-currents. — It is often desirable to know the direc*
tion as well as the velocity of flow beneath the surface. This
is of especial importance in surveys for the improvement of the
mouths of tidal rivers and the adjacent harbors. For this pur-
pose the Ritchie-Haskell* Direction-Current Meter, Fig. 80, is
well adapted.
Fig. 80,
With this meter the observer is enabled to determine,
simultaneously, on dials before him, the direction and velocity
of any current.
The Direction Part. — The central chamber of the meter is a
ConapasSy whose needle is free to assume the magnetic meridian
• Invented by E. S. Ritchie and E. E. Haskell, the latter of the U. S. Coast
aad Geodetic Survey. Manufactured by E. S Ritchie & Sons, Brookline, Mass.
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332 SURVEYING.
at all times. This chamber is filled with oil, giving stability to
and preventing rust of the needle and other mechanism. An
expansion bag compensates changes in temperature and estab-
lishes equilibrium between inside and outside of chamber when
immersed.
By the use of an electric current, the angle to the nearest
degree between the direction of the current and the magnetic
needle or meridian can be measured on the dial shown in the
cut.
The Velocity Wheel. — This is of the propeller type, conical
in form to prevent the catching of weeds, grass or roots. A
pitch has been given to the flukes that makes it extremely sen-
sitive to low velocities, and it is said to register accurately as
low as 0.20 of a foot per second. The electric connections are
made on the inside through the hub of the wheel, so that they
cannot be deranged, and are positive in action. It will record
on any counting register or a chronograph.
This instrument has been used on the U. S. Coast Survey
with very satisfactory results. It would appear to possess
advantages over all other forms of current meter, even when
used without the direction part.
256. The Flow of Water over Weirs**— The most ac-
curate mode of measuring the flow through small open channels
IS by means of weirs. There are three kinds of weirs with
which the engineer may have to deal in measuring the flow of
water, — namely, sharp-crested weirs, wide-crested weirs, and
submerged weirs.
A sharp^rested weir is one which is entirely cleared by the
water in passing over it, as in Fig. 81. A wide crest is shown
* The results given in this and the following article have been mostly taken .
from a paper by Fteley and Stearns before the Am. Soc. Civ. Engrs., vol. xii.
(1883), describing experiments made in connection with the new Sudbury River
Conduit, Boston, Mass. The paper was awarded the Norman medal of that
society.
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HYDKOGRAPHIC SURVEYING.
333
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334 SURVEYING.
in Fig. 82, and its effect in increasing the depth on the weir
for a given discharge. If the crest has a width equal to the
line ab in Fig. 81, then the depth on the weir is unaffected,
while if it has a less width, as in Fig. 84, and if the air has not
free access to the intervening space beneath^ the water will soon
fill this space, and the tendency to vacuum here will depress
the overflowing sheet of water, thus diminishing the depth on
the weir for a given flow. The dotted lines in Fig. 84 are
Fig. 85.
those of normal flow, the full lines being the new positions
assumed as a result of the partial vacuum below.
A submerged weir is one at which the level of the water
below the weir is above its crest, there being, however, a certain
definite fall in passing the weir, as shown in Fig. 85. Here
h = d — d^ is the fall in passing the weir.
Velocity of Approach, — This is the velocity of the surface-
water towards the weir at a distance above the weir equal
to about two and one half times the height of the weir above
the bottom of the channel.
End Contractions, — These are the narrowing effects of the
lateral flow at the ends of the weir. If this lateral component
of the flow is shut off by a plank extending several feet up
stream and from the water's surface to several inches below
the top of the weir, then there is no end contraction. This
arrangement gives more accurate results, as the correction for
end contraction involves some uncertainties.
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HYDROGRAPHIC SURVEYING,
?3S
Depth of Water on the Weir, — This is the principal function
of the discharge ; it is the difference of eleva-
tion between the top of the weir and the surface
of the water at a distance above the weir equal to
about 2\ times the height of the weir above the
bottom of the channel. Evidently this is a quan-
tity which cannot be directly measured. The
best way of measuring this quantity is as follows:
At a convenient point arrange a closed vertical
box which connects by a free opening with the
channel at about mid-depth at a point some six
feet above the weir. The water will then stand
in this box at its normal elevation, unaffected by
the slope towards the weir. The elevation of
this water-surface is determined by means of a HI j
hook-gauge^ Fig. 86, which consists of a metallic
point turned upwards and made adjustable in
height by means of a thumb-screw. When the
point of the hook comes to the surface of the
water it causes a distorted reflection. The eleva-
tion of the water-surface can be found in this
way with extreme accuracy. The difference of V
elevation between the point of the hook and the LJ
crest of the weir can then be determined with a i^'^- ^^
level and rod. This difference is H \x\ the following formulae.
257. Formulae and Corrections. — For a simple sharp-
crested weir, without end contractions and with no velocity of
approach, the discharge in cubic feet per second is
0 = 3.3iZ//^i + o.oo7Z,
(0
where L is the length of the weir and H the depth of water
upon it, both measured in feet. The weir must have a level
crest and vertical ends ; it should be in a dam vertical on its
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336 SURVEYING.
up-stream side ; the water on the down-stream side may stand
even with the crest of the weir if it has considerable depth.
The error is not more than one per cent when the water on the
down-stream side covers fifteen per cent of the weir area, pro
vided//'is then taken as the difference in elevation of the
water-surface above and below the weir. In this case two*
hook gauges would be needed. The crest of the weir should
be at a height above the bottom of the channel on the up-
stream side equal to at least twice the depth on the weir, to
allow for complete vertical contraction.
The following corrections apply to their respective condi^
tions :
For the velocity of approach^ the depth on the weir, H in
equation (i), is to be increased by 1.5 A, where there is no end
contraction, h being the head due to the velocity, or A = — .
At sea-level this correction becomes
1.5^
C = — — = 0.02342^ * (2)
This is to be added to H in equation (i), v being measured in
feet per second.
Where there is end contraction, the correction is
^ 2.05^^
C = -^ = o.032z;« (3)
For end contraction, the length of the weir, L in equation
(l), is to be shortened by o.i ^ for each such contraction. This
is a mean value, although it varies from o.oyH to 0A2H for
different depths on the weir varying from i to 0.3 foot, the
smaller correction applying to the greater depth on the weir.
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HYDR0GRAPH2C SURVEYING. 337
For wide crests the correction to the depth on the weir is
sometimes positive and sometimes negative, as shown in fig-
ures 82 and 84. The following correction is derived from care-
ful experiments :
C = 0.2016 fy +0.2146Z1;* — o.\%^6w, ... (4)
where
C is the correction to be added algebraically to the depth
on the wide crest to obtain the depth on a sharp crest
which will pass an equal volume of water ;
w is the width of the crest ;
y is the difference between 0.807W and the depth on the
crest.
If the crest is narrower than the line ab. Fig. 81, then this
correction is not to be applied unless the water adheres to the
weir as in Fig. 84.
Up-stream edge of the weir rounded. If the up-stream edge
of the weir is a small quarter-circle, add seven tenths of its ra-
dius to the depth on the weir before applying the general weir
formula.
Submerged weir. When the water on the down-stream
side rises above the level of the crest, use the formula for a
submerged weir, which is
Q = cl[c
dT^
{^+1)^' (5)
where
Q is the discharge in cubic feet per second ;
c is to be taken from the following table, its value varying
with -J ;
a
/is the length of the weir in feet;
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338
SURVRYIh'G,
d is the depth on the weir in feet, measured from still
water on the up-stream side ;
d! is the depth to which the weir is submerged, measured
from still water on the down-stream side; „^ ^
h is the fall and equals d— d'.
The value of d may be corrected for velocity of approach
by formulas (2) and (3). There is no known correction for the
velocity of discharge below the weir, and hence the formula
can only be used for a channel of large capacity below, as com-
pared with the discharge, so that the velocity here will be small
The following are the experimental values of c\
d'
c.
d*
d'
c.
1 ''^
d'
c.
d*
d'
c.
O.OI
3.330
0.25
3.249
0.55
3.100
0.85
3.150
.05
3.360
.30
3.214
.60
1
3.092
.90
3.190
.08
3.372
.35
3.182
, .65
3.089
.95
3.247
.10
3.365
.40
3.155
.70
3.092
1. 00
3.360
.15
3.327
.45
3. 131
.75
3.102
.20
3.286
.50
3. "3
.80
3.122
d'
This table is inapplicable to values of -\ less than 0.08, un-
less the air has free access to the space underneath the sheet.
The method of measuring discharge by means of sub-
merged weirs is adapted to channels having very small slope.
A fall as low as one half inch will give reliable results if it is
accurately measured.
258. The Miner's Inch. — This is an arbitrary standard
both as to method and as to volume of water discharged. It
rests on the false assumption that the volume discharged is
proportional to the area of the orifice under a constant head
above the top of the orifice. Its use grew out of the necessities
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HYDROGRAPHIC SURVEYING 339
of frontier life in the mining regions of the West, and should
now be discarded in favor of absolute units. The miner's inch is
the quantity of water that will flow through an orifice one inch
square^ under a head of from four to twelve inches, according to
geographical locality. Even if the head above the top of the
orifice be fixed, and a flow of 144 miner's inches be required,
the volume obtained would be 3.3, 4.2, or 4.7 cubic feet per
second, according as there were 144 holes each one inch square, -
one opening one inch deep and 144 inches long, or one opening
twelve inches square, the tops of all the openings being five
inches below the surface of the water. This simply illustrates
the unreliable nature of such a unit. In some localities the
following standard has been adopted : An aperture twelve
inches high by twelve and three-quarter inches wide through
one one- and one-half-inch plank, with top of opening six inches
below the water-surface, is said to discharge two hundred
miner's inches. By this standard the miner's inch is 1.5 cubic
feet per minute, or 2160 cubic feet in twenty-four hours.
Other standards vary from 1.39 to 1.78 cubic feet per minute.*
When the miner's inch can only be defined as a certain num-
ber of cubic feet per minute, it is evidently no longer of ser-
vice and should be abandoned. The method by weirs is more
accurate, and could almost always be substituted for the
method by orifices.
259. The Flow of Water in Open Channels. — For more
than a century hydraulic engineers have labored to find a fixed
relation between the slope and cross-section of a running stream
and the resulting mean velocity. If such a relation could be
found, then the discharge of any stream could be obtained at
a minimum cost. It is now known that there is no such fixed
relation. There certainly is a relation between the bed of a
stream for a considerable distance above and below the section,
♦ See Bowie's ** Hydraulic Mining," p. 126 (John Wiley & Sons, New York).
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340 SURVEYING,
the surface slope, and tlie resulting velocity at the section;
but as no two streams have similar beds, nor the same stream
in different portions of its length, and since the bed character-
istics are difficult to determine, and, furthermore, are constantly
changing in channels in earth, the function of bed cannot be
incorporated into a formula to any advantage except for chan-
nels of strictly uniform and constant bed, in which case the
cross-section would sufficiently indicate the bed. Again, the
slope cannot be profitably introduced into a velocity formula
except where it is uniform for a considerable distance above
and below the section, for the inertia of the water tends to
produce uniform motion under vaiying slopes, and the effect
is that the velocity at no point corresponds strictly to the
slope across that section. For uniform bed and slope, how-
ever, formulae may be often used to advantage.
Let A = area of cross-section ;
V = velocity in feet per second (= / for one second) ;
p = wetted perimeter ;
r = hydraulic mean radius = -- ;
/
Z
s = surface-slope = sin / = -^ ;
Z = fall per length /;
Q = quantity discharged in one second ;
5 = wetted surface in length / = //;
/ ^ coefficient of friction per unit area of S;
0 = weight of one cubic foot of water = density.
Since the friction varies directly as the density and as the
square of the velocity, we have for the frictional resistance on
the mass covering the area 5,
R==/pSv' , (I)
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HYDROGRAPHIC SURVEYING. 34I
and the work spent in overcoming this resistance in one sec-
ond of time is
K=Rv=fpSv* (2)
If the velocity is constant, which it is assumed to be, then
this is also the measure of the work gravity does on this mass
of water in puUing it through the height h' — h" = Z, which
Work is
K = weight X fall = ZpQ = ZpvA ; . . . (3)
.'. ZpvA = /pSv\ (4)
^=■§^^ (5)
A Z
But S=pl\ -=r; and -^ = sin i = s;
: - — or V = c Vrs, . . c . (6)
where c is an empirical coefficient to be determined. It is evi-
dent that c is mostly a function of the character of the bed,
and that it can, therefore, have no fixed value for all cases.
Equation (6) is what is known as the Chezy formula. The
most successful attempt yet made to give to the coefficient c
a value suitable to all cases of constant flow is that of Kut-
ter.* Kutter\s formula, when reduced to English foot-units, is
* Kutter*s Hydraulic Tables, translated from the German by Jackson, and
published by Spon, London, 1876. A revised and enlarged edition has now (1890)
been edited by Rudolph Hering and J. C. Trautwine, Jr.. and published by John
"Wiley & Son, New York.
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342
SURVEYING.
V = c Vrs =
^ , I.81I , 0.00281
41.6 '\ -]
, / ^ , 0.0028 1 \ n
1 + ^41.6+—^-;-;;^
i^rs, . . (7)
the total coefficient of the radical, in brackets, being the eval-
uation of c in equation (6). Here v, r, and s are the same
as before, and « is a " natural coefficient ** dependent on the
nature of the soil, character of bed, banks, etc. Although it
was the author's intention to make a formula that would be
appUcable even to natural channels, it cannot safely be ap-
plied to such unless they have great uniformity of bed and
slope.
The following values of n are given by Kutter: '
Planed plank,
n = 0.008.
Pure cement.
n = .009.
Sand and cement,
n = .010 to .oil.
Brickwork and ashlar,
n = .012 " .014.
Canvas lining,
n = .015.
Average rubble.
n = .017.
Rammed gravel.
n = .020.
In earth — canals and ditches.
n = .020 to .030,
depending on the reg-
ularity of the cross-
section, freedom from
weeds, etc.
In earth of irregular cross-section,
n = .030 to .040.
For torrential streams.
n = .050.
In the last two cases the results are very uncertain.
ter*s tables are evaluated for ;/ = 0.025, .030, and .035.
Kut.
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nYDROCRAPHlC SVRVEVWG, 343
The greatest objection to the use of this formula is the
labor involved in evaluating the ** c'' coefficient. To facilitate
the use of the formula this coefficient has been evaluated for a
slope of o.ooi in Table* IX. This coefficient changes so
slowly with a change in slope that the error does not exceed
3f per cent if the table be used for all slopes from one in ten
to one in 5280, which is a foot in a mile. These tabular co-
efficients may therefore be used in all cases of ditches, pipe
lines, sewers, etc. The coefficients are seen to change rapidly
for different values of «, so this value must be chosen with
care.
For brick conduits^ such as are used for water-supply and
for sewers, the fcfrmula
zf = I27r^^s^s
was found to represent the experiments on the Boston con-
duit, shown in Figs. 78 and 79. This would correspond to a
variable value of n in Kutter's formula, being nearly 0012
however, as given for brickwork. This conduit is brick-lined.
Table X.* gives maximum discharges of such conduits
as computed by Kutter's formula, n being taken as 0.013.
The results in heavy type include the working part of the
table for sewers. All values above the heavy-faced type cor-
respond to velocities less than three feet per second when the
depth of water is one eighth of the diameter, or when the flow is
one fiftieth the maximum. This is as small a velocity as is con-
sistent with a self-cleansing flow in sewers. All values below the
heavy-faced type correspond to velocities more than fifteen feet
per second when the conduit runs full, and this is as great a ve-
locity as is consistent with safety to the structure. If the velocity
is greater than this, the conduit should be lined with stone.
♦ Taken from a paper by Robt. Moore and Julius Baier \vl Journal of the Asso-
ciation of Engineering Societies ^ vol. v., p. 349. This table may also be used for
tile drains.
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344
SURVEYING.
The maximum flow does not occur when the conduit runs
full, but when the depth is about 93 per cent of the diameter.
A conduit or pipe will therefore not run full except under
considerable pressure or head. The maximum velocity occurs
when the depth is about 81 per cent of the diameter.
The relative mean velocities and discharges of a circular
conduit for varying depths is shown by the following table:
Depth of
Water.
Relative
Velocity.
Relative
Discharge.
Depth of
Water.
Relative
Velocity.
Relative
Discbarge.
.1
.28
.016
.7
.98
.776
.2
.48
.072
.75
•99
.850
.25
.57
.118
.8
.99
.912
.3
.64
.168
.81
1. 00
.924
.4
.76
.302
.9
.98
.992
.5
.86
.450
.93
.96
1. 000
.6
•93
.620
1. 00
.86
.916
260. Cross-sections of Least Resistance. — From equa-
tion (6) of the preceding article it is apparent that for a given
channel the velocity varies as the square-root of the hydraulic
mean radius, r. But r = --, hence for a given area of cross-
/
section the velocity is greater as the wetted perimeter is less.
The form of cross-section having a minimum perimeter for a
given area is the circular, or for an open channel the semicircu-
TtR" ^R
2nR "" 2 '
where R is the radius of the circle. Since it is not always con-
venient to make the cross-section circular in the case of ditches
and canals, it is evident that the more nearly a polygonal
cross-section coincides with the circular form the less will be
the resistance to flow. When a maximum flow is desired for
lar. In both cases the hydraulic mean radius is r =
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HYDROGRAPHIC SURVEYING. 345
a given slope and cross-section, therefore, the shape should
conform as nearly as possible to that of a semicircle. To do
this, construct a semicircle to scale of the required area of
cross-section. Draw tangents for the
sides of the section having the de-
sired slope and join these by another
tangent line at bottom, as in Fig.
87. This gives a little larger section-
al area, but some allowance should fig. 87.
be made for accumulations in the
channel. If the slope is very great and it is desirable to re-
duce the velocity of flow, it may be done by making the
channel wide and shallow.
261. Sediment-observations. — It is often necessary in sur-
veys of sediment-bearing streams to determine the amount of
silt carried by the water in suspension. The work consists of
three operations, namely : (i) obtaining the samples of water ;
(2) weighing or measuring out a specific portion of each, mix-
ing these in sample jars according to some system, and setting
away to settle ; (3) siphoning off the clear water, filtering, and
weighing the sediment. Sometimes a fourth operation is re-
quired, which is to examine the sediment by a microscope on
a graduated glass plate, and estimate the percentages of differ-
ent-sized grains. The sedimentary matter carried in suspen-
sion may be divided into two general classes, — that in continu-
ous suspension, and that in discontinuous suspension. The
former is composed of very fine particles of clay and mud
whose specific gravity is about unity, so that any sh'ght dis-
turbance of the water will prevent its deposition. This once
taken up by a running stream is carried to its mouth or caught
in stagnant places by the way. The matter in discontinuous
suspension consists of sand, more or less fine according to the
velocity and agitation of the current. This matter is con-
stantly falling towards the bottom and is only prevented by the
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346
PURVEY INC,
K.^^-
lil
violent motions of the medium in which they are suspended.
These particles are constantly being picked up where the ve-
locity is greater, and dropped again where the velocity is less.
A natural channel will therefore carry about the same per-
centage of fine or continuous matter between
/I two consecutive tributaries, but of the coarser
1-^ material there will be no uniformity whatever in
successive sections in this same stretch of river.
In natural channels there are always alternate
engorged and enlarged sections for any particu-
lar stage of river, and the positions of these en-
gorgements and enlargements are different for
different stages. In fact, the engorged sections
at high water are usually the enlarged sections
at low water, and vice versa. If the bed is fria-
ble the engorged section is always enlarging, and
the enlarged section is constantly filling as a
result of the discontinuous movement of sedi-
mentary matter. The cause of these relative
changes of position of engorged and enlarged
sections is the great variation in width.*
It is the discontinuous sediment which is of
principal significance to the engineer, for this
is the material from which sand-bars are formed
which obstruct navigation, and it is also the ma-
terial from which he builds his great contraction
works behind his permeable dikes. The water
being partially checked behind these dikes at once drops the
heavier sediment, and so artificial banks are rapidly formed.
The continuous sediment is of little consequence to the engi-
neer.
Fig. 88.
* See paper by the author entitled " Three Problems in River Physics." be-
fore the American Association for the Advancement of Science, Philadelphia
meeting, 1884.
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HYDROGRAPHIC SURVEYING. 347
262* Collecting the Specimens of Water. — It is neces-
sary to take samples of water from various parts of the cross-
section in order to obtain a fair average. Surface and bottom
specimens should always be taken, and if great exactness is
required specimens should also be taken at mid-depth. One
of each of these* should be taken at two or three points on the
cross-section. A full set of specimens is collected once or
twice a day. A special apparatus is required for obtaining
samples from points beneath the surface. The requirements
of such an apparatus are very well satisfied by the device
shown in Fig. 88, which the author designed and used very
successfully in a hydrographic survey of the Mississippi River
at Helena, Ark., in 1879.* Cis a galvanized iron or copper
cup ; /an iron bar one inch square; L a mass of lead moulded
on the bar at bottom ; B the bottom cup for bringing to
the surface a specimen of the bottom, / being a leather cover;
W^the springing wire by which the lids ^ ^ are released and
drawn together by the rubber bands b b when the apparatus
strikes the bottom, or when this wire is pulled by an auxil-
iary cord from above; dd adjustable hinges allowing a tight
joint on the rubber packing-disks c c when the lids are closed.
In descending, the lids are open and the water in the can C is
always a fair sample of the water surrounding the apparatus.
When the lids are closed the sample is brought securely to
the surface. The can when closed should be practically water-
tight ; if it leaks at bottom some of the heavier sediment is
likely to escape, for it settles very quickly. The bottom speci-
men should be taken about a foot above the bottom to avoid
getting an undue portion of sand which is at once stirred up
by the apparatus striking the bottom.
263. Measuring out the Samples. — A given portion of each
specimen by measure or by weight is selected for deposition.
* Sec Report of Chief of Engrs., U. S. A., 1879, vol. iii., p. 1963.
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348 SURVEYING,
Great care must be exercised in obtaining the sample volume.
It cannot be poured off, even after violent shaking, for the
heavy sand falls rapidly to the bottom. A good way is to
draw it from the vessel by an aperture in its side while the
water is stirred within ; greater accuracy can be attained by
weighing the sample of water than by measuring it. All the
samples of a given kind are then put together in one jar, which
is properly labelled, and set away to settle. Thus, all the sur-
face samples are put into one jar, the mid-depth samples in
another. The Mississippi and the Missouri River water re-
quires about ten days* settling to become clear.
264. Siphoning ofT, Filtering, and Weighing the Sedi-
ment,— When the water has become quite clear it is carefully
siphoned off, and the residue is filtered through fine filter paper
(Munkteirs is best). Two papers are cut and made of exactly
the same weight. One is used for filtering and the dupHcate
laid aside. The filter-paper containing the sediment and also
its duplicate are then driefd in an oven at a temperature not
higher than 180°. When quite dry the sediment paper is put
in one pan of the balance, and the duplicate in the other and
weights added to balance. The sum of the weights is ^he
weight of the sediment. This divided by the weight of the
sample of water, usually expressed by a vulgar fraction whose
numerator is one, is the proportionate quantity sought.
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CHAPTER XL*
MINING SURVEYING.
INTRODUCTORY.
The subject of mining surveying as treated in this chapter
will include the surveys necessary in obtaining title to the
mineral lands of the United States, the surveys for the location
of the surface improvements and boundary lines, as well as the
underground surveying necessary in making connections, lay-
ing out work, determining the relation of the underground
workings to the surface lines, points, etc., and measuring the
ore removed or still in the mine. For definitions of mining
terms, see p. 399.
SURVEYING MINING CLAIMS.
265. Title to Mining Claims. — The public lands of the
United States have been divided into mineral and agricultural
lands, the former being subject to appropriation under the
statutes as lode claims, placer claims, mill sites, and tunnel
sites. The title becomes initiate by discovery ; this is followed
by location and record which completes the possessory title.
This title can be afterward, except in the case of tunnel sites,
perfected by patent proceedings so that a fee simple deed to
the claim is obtained from the government.
♦This chapter has been rewritten for the fifteenth edition (1900) by
Prof. Robert S. Stockton, E.M.» of the Colorado State School of Mines,
and by Mr. Edward P. Arthur, Jr., E.M., U. S. Deput Mineral Surveyor,
of Cripple Creek, Col.
349
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35^ SURVEYING
^^ Fundamental Conditions governing the Location of Mining
Claims. — In the case of a lode claim there are five essentials to
a valid location : (i) discovery of lode or vein ; (2) posting of
location notice ; (3) sinking of discovery shaft ; (4) marking
the boundaries of claim ; (5) the making and filing of the loca-
tion certificate. Lode claims are limited by the U. S. statutes
(R. S. 2320) to 1500 ft. along the vein or lode and to 600 ft. in
width, i.e., 300 ft. on each side of vein. The end lines must be
parallel, and the side lines usually are. The different States,
and originally the different districts, had the right to limit the
width, at the same time keeping within the congressional
limits. In Colorado the width has been reduced to 300 ft.,
except in the counties of Gilpin, Summit, Clear Creek, and
Boulder, where it is only 150 ft. North Dakota and South
Dakota fix the width at 300 ft., allowing the counties to reduce
this within the congressional limits. All the other States have
adopted the width of 600 ft. except where limited by the old
district rules.
In locating a placer claim^ the discovery of auriferous gravel
or any of the substances designated by law as subject to placer
location is followed by the posting of a location notice, marking
the boundaries, and making and filing the location certificate.
One person or corporation may appropriate 20 acres, or an
association of at least 8 persons may secure 160 acres, of placer
ground. The ground appropriated may conform to the govern-
ment land subdivisions if on surveyed lands, or it may be in
any shape bounded by straight lines.
A mill site may be located on non-mineral land by posting
a location notice, marking the boundaries, and filing a location
certificate. The area of a mill site is limited by the U. S.
statutes to 5 acres, and may be further limited by district rule.
It may be taken in any desired form bounded by straight lines.
A very convenient size is 726 ft. by 3CXD ft.
A tunnel-site location appropriates the right of way for a
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MINING SURVEYING. 35 1
tunnel 3000 ft. long, and may in addition appropriate ground
for a dump up to an area 250 ft. square. The location is
accomplished by posting a notice, marking the line of tunnel
and the corners of the dump area by stakes properly inscribed,
and filing the location certificate, which must in this case be
acknowledged before a notary public. The tunnel site differs
from the other locations mentioned in that it is not subject to
patent. The veins cut by the tunnel are located as lode claims
and may be patented as such, the title dating back to the loca-
tion of the tunnel site. Under recent decisions the tunnel site
practically withdraws from the public domain all the lodes
within an area of 3000 ft. square that were not located prior
to the tunnel site.
The possessoiy title to a tunnel site involves what the law
calls continuous working. This has been held to mean no
cessation of work for more than six months.
266. Location Surveys. — A surveyor is generally called
upon to lay out the claim, mark the boundaries, and fill in the
location certificate ready for the owner to file with the county
or district recorder. The law, however, allows the location
survey to be made by the miner himself without an instrument,
i.e., the boundaries may be laid down very roughly, if with
honest intent, and hold in law, but locators have found it
profitable to have the claims surveyed to avoid the losses result-
ing from gross blunders so often made when unskilled persons
have attempted to lay out the claim. The survey should be
made with sufficient care so that when the ground is surveyed
for patent the corners will be identical with, or come within,
the location corners and also fulfil other necessary conditions,
as will be shown under the discussion of patent work.
This class of surveying will be discussed under four heads
viz. : lode claims, placer claims, mill sites, and tunnel sites.
267. Surveying Lode Claims. — The simplest form of a
lode claim resulting from the conditions laid down in Art. 265
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352 SURVEYING.
IS a rectangle 300 X 1 500 ft. The two things which should
govern the exact direction of the lines are the direction of the
vein and the position of the discovery. The usual method is
to set up the instrument at or near the point of discovery.
The direction of the vein is determined by the outcrop or any
other available data, but the vein line is often run so as to take
in any vacant ground. The line is then run out, its bearing
being taken from the true or magnetic meridian. The distances
are measured with a steel tape either horizontally or by taking
the slope distance and slope angle and reducing to the hori-
zontal, this work being done with about the same accuracy as
is used in ordinary land surveying. The lode or vein line is
run both ways from the discovery, the only condition being
that the total distance is not more than 1 500 ft. From the
ends of the lode line, in this case the centre line, and at right
angles to it, and distant 150 ft., are set the corner stakes.
From the centre of the claim the side centre stakes are also
set. In Fig. 89 is shown a diagram of the claim and also the
N0.1.
N.t.eo<t
AMOWLOOC
N0.«.
•.B.OOR.
AmiOWLOO
1
NO
N.W.
COR.
WKST tioe OCNTtR
NO
•.W.<
a.
'Jan.
Fig. 89.
markings on the corner stakes. The dotted lines show those
actually run on the ground. The comer stakes should be 3 or
4 in. square, and 3^ ft. long, set in the ground or in a mound
of stone. They should also be blazed and marked on a side
toward the claim with the number of the corner and the name
of the claim.
One or more of the corners should be tied in by bearing
and distance to some permanent objects, as a government
survey corner, or the corners of an official survey of some
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MINING SURVEYING.
353
Other mining claim, or, failing these, to bearing trees and rocks.
A practice is in vogue of connecting claims to mountain peaks
by bearing only, but it is not to be recommended except as a
last resort. It may be very often advisable to mention in the
description of the claim the name of the mountain on which it
lies, or the names of adjacent roads, streams, etc., as a help to
identification.
Another simple form of a lode claim is where the end lines
are not perpendicular to the lode line and side lines, the result-
ing figure being a parallelogram as shown in Fig. 89^. This
NOrrH tlK CCNTER
M0.1.
N.C.COII.
TWILKIHT LOM
-OISCBMAFT
t.^
NO.*.
•.W.COR.
TWILMHTLOOC
SOUTH SIOC OfiNTCR
NO. 2.
•.CCOfl.
TWILKIHT LOM
Fig. 89«.
condition generally results from the claim abutting on a fixed
line.
The surveying is just the same as in the first case except
that the distance measured from the lode to the side lines along
the end lines, or a line parallel to them at the middle point, is
greater than 150 ft. and is found by dividing 150 by sin a,
where a is the angle between the end line and lode line as
shown in the figure.
Fig. 89^.
If the vein bends, the lode line is made to follow it, thus
forming a crooked claim. Two forms are shown in Figs. 89^
and 89c.
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354
SURVEYIXG,
In Fig. 89^ let /? represent the angle between the two direc-
tions of the lode line, then the line from Cor. No. 3 to Cor. No.
6 will bisect this angle if the side lines are parallel and at equal
distances in both parts of the claim. The distance from the
discovery to Cors. Nos. 3 and 6 will be 150 divided by sin \Q,
If the end lines are not fixed as to direction by some outside
condition they may be made parallel to line 3-6, and will by
Fig. Sgr.
the geometric conditions laid down be exactly equal to it.
The lode line is run and corners set as before except that the
side centres may be omitted if the bend is near the centre of
the claim.
In Fig. 89^: let /? represent the angle between the two parts
of the lode line, then the line from Cor. No. 3 to Cor. No. 6
will bisect the angle ^ as in Fig. 89^. Now if one end line, as
line 4-5, is to be perpendicular to the lode line, it will be 300
ft. long and not parallel to the line between 3 and 6, while line
1-2 is parallel to 4-5, but has a length of 300 divided by cos tr.
The changes in the surveying will be made sufficiently clear
by a glance at the figure.
The Location Certificate is generally made out by the
surveyor, printed forms being used with blanks for the name
of the lode, the name of the locator, the date of the location,
the number of feet claimed on each side of the discovery, and
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AflNING SURVEYING, 355
a general description giving the length and the bearing of the
boundary lines and of the tie lines
The location survey for a placer claim or a mill-site does not
differ from an ordinary land survey to establish boundaries.
The corners are like those of lode claims and are marked with
the name of the placer or mill-site and the number of the
corner on the side toward the claim. The tie lines are also
the same as in the lode claims. The location also covers about
the same points and is made out in a manner very similar to
the lode certificates.
As a tunnel site is not subject to patent, but is held by
possessory title only, the survey is properly a little more
accurate than the location surveys. The true meridian should
be determined as in patent work. The survey consists in
running out the line of the tunnel, placing stakes close enough
together so that one can be seen from the other and not
further than 300 or 400 ft. apart as a maximum distance. The
manner of setting stakes varies with the different surveyors,
but the above method seems to be the best practice. These
stakes are marked by the number and the name of the tunnel
site. The dump area is also marked by a stake at each corner.
The location certificate is rather an uncertain document,
several forms being used. A very good form is given in
Morrison's Mining Rights,
268. Patent Surveying. — In order to complete the title to
any mining claim by obtaining a U. S. patent, it is necessary
to have the boundaries carefully surveyed and substantial
monuments set. The monuments are connected with some
permanent object in order to fix the position of the claim.
This work is known ^s patent surveying ^nA can only be under-
taken by a U. S. Deputy Mineral Surveyor holding a com-
mission from the government. This commission may be
obtained by passing an examination under the Surveyor-
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3S6 SURVEYING.
General of the State in which it is desired to practise, and by
filing a bond for $10,000.
Patent work follows the same general plan as location
surveying in the running of the boundaries and setting the
corners, but is done with much more accuracy. In patent
work all conflicting prior official surveys are connected with
the claim under survey, and improvements of all kinds are
located. The following list of apparatus, while not absolutely
necessary for this work, has been found by those familiar with
the best practice to be most convenient. It consists of: (i) A
light mountain transit, mounted on an adjustable tripod, and an
adjustable plumb-bob. The plates and vertical arc or circle of
the transit are 4^ to 5^ inches in diameter and read to minutes.
(2) A steel tape, preferably 500 ft. long, usually graduated
every 5 or 10 ft. (3) A short ribbon tape graduated to
hundredths of a foot, together with a note-book, field-book of
tables, axe, nails, chisel, and timber-scribe. Some surveyors
carry a rod and also some red cloth for marking stations.
A good pick and shovel are necessary for setting corners.
Instrumental Work. — In this kind of surveying the compass-
needle is never used except as a rough check, all angles being
read on the plate circle by means of the verniers. There are
two general methods in vogue in the mining districts of
carrying the meridian from one station to another. These
will be designated as the Azimuth and the Angle methods.
The better method is by azimuths, as described in this book,
or its modification, where the bearings are carried on the
horizontal limb in the same way as the azimuths. It is well
for beginners to check the readings of all important sights at
least. If the instrument has been oriented and a sight taken and
recorded, this can be checked by setting on the back azimuth
reading, and sighting to see if the cross-wires are on the point ;
then loosen the alidade and set on the recorded forward reading,
and again sight to see if the forward point is caught ; if so, the
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MINING SURVEYING. 357
recorded reading is correct. In the angle method the
vernier is always set on zero for the backsight and the angle
read to right or left, as the case may be, to the forward point.
A nice check may then be made by loosening the lower motion
and turning on to the back station, then loosening the alidade
and turning again on to the forward point. The angle now
read should be just twice the original one for a check. This
method is much in use by old railroad men, and although it
has a good number of followers it cannot be recommended.
Care should be taken in reading the vertical angles, as it seems
easier to make a mistake in reading these than in reading the
horizontal angles. With a complete vertical circle the best
check is to reverse the telescope and turn on the point and
read again. With a vertical arc a good check may be had by
resetting the recorded angle on the arc and then sighting to
see if the horizontal wire cuts the point whose angle is being
checked.
Too short a backsight is to be avoided. It is often pos-
sible before leaving a station to obtain a distant foresight or
backsight, and this should be done when practicable.
Stations and Methods of Measuring. — Stations are generally
stakes driven into the ground, the height above the surface
being determined by the nature of the country ; the exact
station-point is marked by the head of an eightpenny or a ten-
penny wire finishing-nail, driven about half way into the stake.
Stations may be taken on fallen logs or tree-stumps by
putting in a nail as before. In winter weather, if a clear sight
can be obtained, a forty-penny spike driven directly into the
frozen ground will make a good station. It is usually a good
plan to put a piece of white paper on the station nail in order
that it may be more easily seen.
In measuring, pins are not used, nor is there any special
effort made to have the tape horizontal, the measurement
being taken from the horizontal axis of the instrument directly
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358 SURVEYING.
to the nail-head, and the vertical angle read to the same point,
thus enabling the horizontal distance to be calculated.
The readings are made to hundredths of a foot. Some
surveyors measure back and forward from each station as a
check, but this is hardly necessary with careful worlc
The distance between stations is limited by the contour of
the country and the length of the tape. With a tape of the
ordinary length the distance between stations would be less
than 500 ft
In making a measurement care should be taken that the
tape is straight and subjected to a pull depending on the dis-
tance that the tape is unsupported. Then get the point on
the tape opposite the axis of the instrument, and with the
small tape measure along from this point toward the zero end
of the long tape to the nearest graduation. This will give the
number of feet and fractions to be added to the reading at the
graduation mark, and the sum is recorded as the slope
distance.
Field-work and Adjustment of Claim. — The field-work is
generally accomplished with the aid of one assistant, who can
ries the forward end of the tape, puts in stations, clears out
the lines, assists in setting corners, etc. In a heavily timbered
country an extra axeman is of advantage. The field-work con-
sists in the determination of the true meridian by astronomical
observation, as explained in Chapter XIV, finding the position
of the corners of the location survey, the discovery shaft, and
all improvements on the claim, whether made by the claimant
or not. The corners of conflicting prior official surveys must
be located, and a tie made to a corner of the Government land
survey, if one exists within two miles, or, if such does not exist,
the tie is taken to a U. S. L. monument as described in the
Manual of Instructions for the Sumey of the Mineral Lands of
the U. 5. prepared by the Commissioner of the General Land
Office. This work will be alluded to hereafter as the Manual
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MINING SURVEYING, 359
of Instructions, It is reprinted in full in Appendix B and must
be studied in connection with all that is written here in regard
to patent work.
Before the deputy can have official authority to survey the
claim the applicant must send his application for an order for
the survey, accompanied by a certified copy of the "location
certificate," to the Surveyor-General, and deposit the required
fees (see Manual of Instructions^ The Surveyor-General will
then issue to the surveyor mentioned an order for survey, des-
ignating the survey number to be used. A copy of the certi-
fied copy of the location certificate is also enclosed.
It is well, however, to tie up the claim before the claimant
makes application for the survey order, and, if the location does
not agree with the description given in the location certificate,
to make out and file an amended certificate giving the correct
description. This may save some time and a little expense.
The Land Office at the present time is very particular and will
allow but slight variation between the field-notes turned in by
the deputy and the location certificate on which the order for
survey was based.
The next step, after tieing up the claim, is to compute or
adjust the lines of the final claim to be used in the application
for the patent.
The conditions are imposed by the fundamental principle
that monuments hold over descriptions and by the law and
Land Office regulations on this subject. These conditions are
summarized as follows :
1. The final claim must lie wholly within the location
stakes.
2. The length along the vein or lode must not exceed
1500 ft.
3. The end lines must be parallel.
4. The distance from lode line to side line must at no point
exceed half of the statutory width.
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3^0 SURVEYING,
5. The lode line must pass through the discovery shaft.
6. The end lines must be free, as explained in the Manual
of Instructions.
7. The bearings must be given from the true meridian.
Having the lines of the final claim computed in accordance
with the above, the patent corners are set as described in the
Manual of Instructions.
The meridian used should never vary more than 2 or 3
minutes from the true meridian. If the variation with some
conflicting survey is more than this, these surveys must be
reported as disagreeing, as provided in the Manual of Instruct
iions.
It IS now the common practice of deputies to find the me-
ridian by direct solar observation, as explained in Art. 103^,
although some still use one of the various solar attachments.
Below is given an actual example of the notes, etc., of the
Jewel lode in the Cripple Creek Mining District in Colorado.
This was selected as a typical example from a large number of
surveys with which the writers have been connected both in
this and other districts. The survey was made with a light
mountain transit and a 500-ft. tape, etc., as described. The
field-work included the following operations : The true meridian
was determined by direct observation, the image of the sun
being thrown on a card held behind the eyepiece. Two sym-
metrical sets of sun observations were taken, one before and
one after noon. A traverse was run to tie up the location
corners of the Jewel lode, and also the corners of patent sur-
veys, Nos. 9689, 8882, 9134, 9122, 9190, 8927, the improve-
ments, and a corner of the Government survey. In this case
the order for survey had been received by the deputy, 10264
being the survey number assigned.
The notes as actually taken in the field are shown on p. 361.
The first section of the notes taken on Jan. 20 represent the
tieing-up survey, as it is called. When this was finish<.*d, the
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MINING SURVEYING.
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362 SURVEYING.
notes were taken to the office, where the slope distances were
reduced to the horizontal, the sun observations calculated, and
the true meridian determined. The corrected azimuths were
inserted in column 3 of the notes, and the horizontal distances
in column 6 as shown. The work was also platted as shown
by Fig. 90, the transit lines as run in the field being dotted and
the claim lines shown unbroken. The bearing and lengths of
the lines of the claim were calculated from the data obtained
from the notes. It will be seen by Fig. 90 that the north end
of the Jewel lode was taken in by Survey No. 9190, Ida, No. i
lode, a patent claim of prior location. This necessitated the
cutting back of the Jewel lode until it had a free end centre
off the Ida No. i lode (as shown in Manual of Instructions^
Calculations were made from which to set the patent corners
to conform with the above requirement. A traverse was cal-
culated from Station i to Corner No. i.
The notes of Feb. 6, p. 361, show the work of setting the
corners. Fig. 91. Cor. No. i was set from Station i, and Cor.
No. 4 was set from Cor. No. i. Cor. No. 2 was identical with
the S.E. corner of the location, and Cor. No. 3 was set
from it.
The method of measuring a given distance on sloping
ground is best done in the following way : Set a point on line
within a foot or two of the given distance, measure the vertical
angle and calculate the horizontal distance. The remaining
small distance is measured either backward or forward hori-
zontally to the required point.
The style and markings of the corners are described in the
Manual of Instructions, With a stone corner the exact point
is marked by a cross chiselled into the stone. Post corners are
preferably more than 4 inches square, the exact point being
marked by a nail driven in.
In making the calculations a seven-place table of logarithms
is largely used, although a six-place table would serve. A
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MINING SURVEYING. 363
traverse table reading to minutes, as Table IV of this volume,
is very convenient. Some surveyors use calculating-machines.
The areas are computed by dividing the figure into
triangles and quadrilaterals or by double meridian distances.
A planimeter is much used for checking the areas.
All calculations sliould be carefully checked and should be
made Jn a computation- or scratch-book so that they can be
kept for future reference.
After the corners are set and the field-work completed the
notes are taken to the office and worked up, i.e., a traverse is
calculated as from Cor. No. i to the S. \ Cor. of Sec. 32 and a
tie calculated from some corner of the Jewel to a corner of
each prior conflicting official survey. The intersections were
calculated so that the distance from a particular corner to the
line intersected could be given as well as the distance from
the intersection to the interior or nearest corner on the inter-
sected line. Closing traverses, including some convenient
lines of conflicting surveys, were calculated in order to see how
the Jewel checked with these surveys. Finally the areas are
computed, the notes written up, and the preliminary plat made
as shown in the Manual of Instructions.
Fig. 90 shows the working plat. It will be seen that on the
preliminary plat (Fig. 91) the claim is considerably shorter
than the original location. Several official surveys that con-
flicted with the location did not do so with the final survey. Of
the shafts tied up two belong to the claim and two were other
improvements. The more northerly shaft was left out when
the claim was cut back.
It will be noticed that the Jewel claim is cut off at the exact
point where the assumed lode of the claim enters the Ida No. i
lode, at 150 ft. from each of the north corners. The present
and better practice is to leave a little free ground for the
assumed lode line to pass through before entering the claim
for which it is cut off, as in this case the distance along the
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3^4
SUKVEYmc.
Fig. 90.— Preliminakv Survky.
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MINING SURVEYING^
36S
SURVEY NUMBER 10264
PUEBLO LAND D18TRICT
H,E.X8EC.6.T.16 8.
9 — 1 l?iyl5»'e. J
N.8»"64'W. 800 C0ft.7r"~^-
Fig. 91.— pRKLiMiNAKY Plat.
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366 SURVEYING.
north end line of the Jewel from its N.W. cor. to the intersec-
tion of the Ida No. i line would be 151 ft. instead of 150 ft., as
It now is.
The field-notes are written on regular blanks furnished by
the Surveyor-General and must follow the form given in the
Manual of Instructions. They include a description of the
claim by bearing and distance, also the intersections and areas,
and the certificate of $500 worth of improvements, if these are
completed at the time. If they are not, the certificate may be
filed at a later date (see Manual of Instructions, sec. 46). The
present Land Office ruling is that the affidavits of labor and
improvements filed subsequent to the notes must show that
the work was completed prior to the expiration of the period
of publication. It also requires that when two or more loca-
tions are embraced in one application, the value of the labor
and improvements must equal $500 for each and every loca-
tion embraced.
It is required to give the distance and bearing along the
lode line on each side of the discovery shaft. This last state-
ment is usually put in the notes just before the statement of
areas, though not shown in the specimen field-notes in the
Manual of Instrustions.
The preliminary plat is generally made on tracing-cloth to
a scale of 200 ft. to i inch, and shows the claim with its con-
flicts and improvements. It must give the bearing and length
of all lines used in the notes. * The notes with the affidavits of
the deputy and his assistants, the copy of the location certifi-
cate received, with the order from the Surveyor-General, and
any report the deputy may have to make, together with the
preliminary plat, are returned to the Surveyor-General's office,
where they are examined and the work checked ; if found
correct, they are approved and the Surveyor-General's certifi-
cate of approval attached.
A copy is then made of the approved notes, attached to
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MINING SURVEYING. 367
which is the Surveyor-Generars certificate that it is a true
copy of the notes, and that the proper amount has been
expended in labor and improvements, in case the affidavit has
been filed. The approved notes, together with a number of
copies of the approved plats one greater than the number of
locations embraced in the claim, are now returned to the
deputy who made the survey. The approved plats are much
the same as the preliminary plat made by the deputy, except
that the Surveyor-General's certificate of approval and of
expenditure is attached.
This properly finishes the work of the surveyor, who now
turns the notes, plats, etc., over to an attorney. The surveyor
should keep a record of all official correspondence and of all
notes returned to the Surveyor-General's office ; also a copy
of all plats.
269. Placer Claims. — The patent survey of a placer claim
is much the same as that of a lode claim except that it is not
limited to such a great extent. Like a lode claim, however,
it must coincide or lie within the location boundaries. The
lines of a placer claim cannot lap over other claims, and the
area in conflict be excluded as with lode claims. All known
lodes must be excluded. Placers are often patented with lodes
under one survey number. Unlike lode claims, the deputy
must submit under oath to the Surveyor-General's office a
description of the soil, vegetation, etc., on the ground included
within the placer claim (see Manual of Instructions). Two dis-
interested persons are also necessary to make a corroborative
affidavit. When placer ground is patented by legal sub-
division no survey is necessary, but a descriptive report must
be made.
270. Mill-sites. — A mill-site may be patented on non-
mineral land not contiguous to vein or lode, i.e., not joining the
end line of a lode claim, nor including within its boundaries
known lodes. A mill-site when located in connection with a
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3th SURVEYING,
lode claim may be patented under the same sur\'ey number as
the lode and without additional development work. It must
not contain more than 5 acres, and is surveyed and the corners
are set at the same time as the lode claim. The patent survey
must not extend over the location lines. One corner is tied
to a corner of the Government survey, and to a corner of the
lode claim with which it is to be patented. A mill-site may be
patented, not connected with a lode claim, if a mill is actually
built or in process of construction.
271. Amended Surveys. — These surveys are ordered by
special instructions from the General Land Office, and no
rule can be given for the work (see Manual of Instructions),
A very common cause for ordering these surveys is that claims
applying for patent after the survey is made lose the ground
on which one or both end lines are situated. Therefore, after
all suits are settled and before a patent is issued, an amended
survey is ordered to cut back the end lines to free ground.
The amended survey must keep within the original survey.
The survey retains the same number, but must be mentioned
in the notes, etc., as an amended survey ; as, " Sur. No. 9463
Am." Amended surveys are also ordered when a material
error has been made in the original patent survey and for a
variety of other causes.
272. Adverse Surveys. — When the owner of a mining claim
applies for a patent it often happens that ground is included
that is claimed by other parties as belonging to another claim.
These parties must file a protest in the local land office in order
to protect themselves against the issuance of the patent to the
conflicting ground. The protest is known as " an adverse,"
and must be accompanied by a plat showing the conflict
claimed to exist and a description of the conflict by bearing
and distance. It is required that this plat and survey be made
by a U.S. Deputy Mineral Surveyor under oath, who also makes
affidavit as to the labor and improvements on the adversing
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MINING SURVEYING, 3^
claim. The field-work on an adverse survey is much the same
as the tieing-up survey in patent work, except that it is not
necessary to connect with a corner of the Government survey,
nor to tie in the corners of any conflicting official survey other
than those directly concerned. The adverse plat must show
the whole of the adversing claim and that part of the adversed
claim necessary, also the position of all improvements.
The conflict between the claims is generally colored on the
plat. The description should begin at a corner of the adversed
claim or at a point connected by bearing and distance to such
a corner. The area of the conflict is given. If the adversing
claim has also been surveyed for a patent, no field-work will be
necessary in making the adverse plat.
UNDERGROUND SURVEYING.
273. Underground Surveying has for its object the deter-
mination of the position of the various workings of a mine
with relation to themselves and to the surface boundaries and
improvements, or with relation to a system of reference planes.
The information gathered by means of the survey is used to
determine the proximity of property lines, the neighborhood
of other workings, to give points for running connections, to
locate and map any geological features that are important, to
calculate the ore stoped out or the probable amount in sight,
etc. All of these uses are not necessarily required of one
survey, but it is true that no mining property of any magni-
tude can be worked systematically and intelligently without
surveys. The survey is shown by means of one or more maps.
These maps include a plan, and generally the projection of the
workings on two vertical planes at right angles to each other ;
also, in many cases, sections at particular points. The survey is
generally begun by determining the true meridian, tieing in the
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370 BURV EYING.
boundary monuments and surface workings, and establishing
points near the entrance to the mine. The meridian must
then be carried into the shaft inch'ne, adit, or tunnel, and the
main parts of the mine. This is the most particular part of
underground surveying, as the special difficulties encountered
make great care necessary to avoid serious error. Having the
meridian underground the workings are run out if a general
map is wanted, or if some special problem is to be solved the
data necessary can be obtained.
274. Instruments. — For this work a light mountain transit
similar to the one described for patent work is generally used.
It has, in addition to the points mentioned, an auxiliary tele-
scope on either the top or the side of the main telescope, for
vertical or highly inclined sightings, and for careful work
should have either a complete vertical circle or else have a re-
version level-bubble attached to the telescope. In some cases
for the most particular work, an instrument with eccentric
bearings and striding levels may be advisable. Where long
sights can be taken, a yyy- or soo-ft. tape, with a short one for
reading the fractions, may be used. For short sights, which
are the majority of those taken underground under ordinary
circumstances, a loo-ft. ribbon tape, divided to hundredths,
will be most advantageous. Besides a variety of things to be
found at the mine, the surveyor will need wire for plumbing
shafts, heavy plumb-bobs, extra plumb-bobs for station sights,
screw-eyes, nails, an axe, note-book, pocket-book of tables, etc.
Figures 92 and 92^ show the two types of mining transits that
are most used.
The Use of Top and Side Telescope, — In underground work
we wish to find the bearings of the lines, their lengths, and the
differences of elevation between the stations. The survey is
then a traverse to find the three coordinates of the points
occupied. When the sights become so steep that they cannot
be taken with the main telescope on account of the interfer-
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MINING SURVEYING, 371
ence with the plate, then, unless the instrument i*" provided
with eccentric bearings, either a top or a side telescope is a
necessity. In using the top telescope, it must be adjusted so
that its line of sight is parallel to the line of sight in the main
telescope. If the Saegmueller type of top telescope (Fig. 20,
p. loi) is used, it is adjusted as for solar work, and it is only
necessary to level both telescopes and bring the vertical wire of
the top telescope into the same vertical plane as the vertical
wire of the main telescope. This is done by sighting on a
point some distance away with the main telescope and swing-
ing down till the point comes into view in the top telescope,
then move the top telescope by its lateral tangent screws
until the cross-wires cut the point sighted. The instrument is
now ready for use, but has the disadvantage of low power in
case of the Saegmueller solar form, and is less stable under
trying circumstances than a telescope mounted directly on the
axis.
The adjustable telescope is mounted on a central axis with
trivet-plates and adjusting-screws. With this form the line of
sight is adjusted to the optical centre by rotating in wyes as
with a wye level. In this case wooden wyes may be made to
serve. When this has been done the vertical wires of the two
telescopes are brought into coincidence as above, using the
lateral tangent motion of the top telescope. Then to bring
the horizontal wire of the top telescope to a position such that
the two lines of sight are parallel, measure the vertical distance
between the telescopes and draw two horizontal lines at this
distance apart on a piece of white paper. Sight the horizontal
wire of the lower telescope on the lower line on the paper at a
distance of 50 ft. or more from the instrument ; then bring
horizontal wire of top telescope to coincide with upper mark
by means of the trivet adjusting-screws. This form permits of
very accurate work, ranking next to the special mining transits
with eccentric bearings.
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372
SURVEY.
Pic. 93,
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MINING SURVEY.
373
Fig. 920.
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374
SURVEYING.
The ordinary form of top telescope (Fig. 92^7) is fixed
rigidly in place so that the adjustment given above must be
made by moving the cross-wires. The rigid type of top
telescope is mounted by some makers on two pillars fixed to
the main telescope.
If the measurement is taken from the axis of the top tele-
scope to the next station as shown in Fig. 93, the reductions
are made as follows : Let d equal the slope distance ; r, the
distance between the telescopes ; a, the angle of depression or
elevation : then the horizontal distance is equal to d cos a -j-
r sin a. The difference in elevation is equal to rf sin a T
r cos a ip H.I.
^55^^
Pig. 93.
Pig. 94.
If the top telescope is mounted on standards, or if for some
other reason it is desirable to measure from the horizontal
axis of the main telescope, the reductions are as follows :
In Fig. 94 let d equal the slope distance ; r, the distance be-
tween the telescopes ; and a, the angle of depression or eleva-
tion read. The real angle of depression or elevation of the
measured line is a', a' being equal to or ± sin~M -^j, taking
minus sign for correction to angle of depression, and plus for
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MINING SURVEYING, 375
angle of elevation read. Then horizontal distance is equal to
d cos a'. Difference in elevation is equal to d sin a' ip H.I.
In using a side telescope the lines of sight in the two tele-
scopes should be made parallel in a manner very similar to
that used for the top telescope. The side telescope screws
firmly to the end of the horizontal axis of the transit; it is
generally -provided with tangent-screws for movement in a
vertical plane, and may have adjusting-screws for lateral move-
ment ; this would be called an adjustable side telescope.
To use the adjustable side telescope, first adjust the line of
sight to the line of collimation, using the method by means of
wyes as before, then adjust the horizontal wire by sighting a
point 2CX) or 300 ft. away through the main telescope, swinging
on the vertical axis of the transit until the point comes into
view in the side telescope, and moving this telescope up or
down by tangent screws until the horizontal wire cuts the
point. For the vertical wire measure the distance between
the telescopes and draw vertical lines at this distance apart on
a paper set a distance of 50 or 100 ft. from the instrument.
Now bring the vertical wire of the main telescope to coincide
with one line, and then bring the vertical wire of the side tele-
scope to coincide with the second mark by means of its adjust-
ing-screws. In case of a non-adjusting side telescope make
the correction by moving the reticule. In a very common
form the vertical wire must be brought to the proper position
by means of the movement of the reticule, and the horizontal
wire is adjusted by means of tangent-screws moving the side
telescope. The measurements are now made directly from the
horizontal axis of the instrument to next station and the H.I.
measured. Then the horizontal distance is equal to the slope
distance times the cosine of the angle of elevation or depression,
and the difference in elevation is equal to the sine of the angle
of elevation or depression times the slope distance, plus or
minus the H.I. When the side telescope is used and only one
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376
SURVEYING,
forward sight taken to each station, the azimuth read must be
corrected by an angle whose sine is the distance between the
telescopes divided by the horizontal distance between the sta-
tions. The correction can be obviated by sighting to a double
rod, the two parts being separated by a distance equal to the
eccentricity of the side telescope, but this is not recom-
mended.
It is to be noted that if both the fore and back sights on
any course be taken with the side telescope, the instrument
remains properly oriented at each succeeding new station.
Consequently the field-notes need show no correction, but \Xi
the office those sights taken with the side telescope must have
their azimuths increased or diminished according as the side
telescope is on the right- or left-hand side.
One of the great advantages of the side telescope is that
by reversion the errors in adjustment of side telescope and
transit can be eliminated, provided the instrument is carefully
Fig. 95.
leveled. To accomplish this two sights and readings are taken
for each forward and backward sight, as follows :
The instrument being at station A, Fig. 95, and the tel-
escope direct, station B is sighted with the back azimuth
properly set off and the forward azimuth Z\ and the vertical
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MINING SURVEYING, 377
angle read to station C. Then with telescope reversed station
B is sighted, using the same back azimuth and the azimuth Z'* ^
and the vertical angle again read to station C.
From Fig. 94 it is plain that the average of the two azi-
muths read (the second reading being corrected* by 180° as
telescope was reversed) will be the true azimuth Z between
the stations, thus eliminating the effect of the side telescope
if the latter maintains a fixed relation to the main telescope ;
also eliminating the effect of an inclined horizontal telescope
axis, which is such an important factor where steep sights are
taken. The average of the two vertical angles read will give
the true vertical angle. For an absolute check repeat the
whole operation.
274a. Stations. — Underground stations may consist of
overhead or floor stations, and may be either permanent or
temporary. When it is advisable to establish a station in a tie
or in a stake or plug driven in the floor, the head of a wire
nail is good. For a permanent station in a rock floor a hole
3 to 6 inches deep is drilled and a wooden plug set, in which a
nail can be driven. Some writers mention the use of a f "
boiler-rivet sunk in a hole y diam. and ij" deep. The
rivet is split for an inch and has a wedge started in the split ;
this wedge strikes the bottom of the hole first, spreads the
rivet, and holds it firmly in solid rock.
For permanent overhead stations screw-eyes set into caps
or stulls. from which a plumb-line can be suspended, are the
most convenient; where timbers are not available, a hole
is drilled in the rock from 3 to 6 inches, and a wooden
plug driven in and the screw-eye set as before. Horseshoe-
nails with a hole punched in the head have been used in place
of screw-eyes, but it is an unnecessary trouble to have them
made. A three-inch wire nail, bent in the form of a staple,
makes an excellent station, as the sharp corner allows the
plumb*bob string to always take the same position.
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378 SURVEYING.
In a large mine the stations should be carefully numbered.
If the mine is worked by levels, a good system is to number
stations on ist level loi, 102, etc., those on 2d level 201,
202, etc., and so on for all the levels, thus avoiding mistakes
and saving time in hunting up notes. The numbers may be
painted or scribed on the adjacent timbers or rock. A very
reliable way is to use round-headed nails, as • J i for 324, a
•
washer serving for zero. The very best method, however, is
to use brass or zinc tags marked with stamps. This tag is
nailed to the plug or timber according to the character of the
station.
It is very important in extensive workings to have the
stations legibly and systematically numbered and referenced ;
the latter precaution is especially necessary where the station
is in a timber, as witness the mishap of the man who ran a con-
nection from a station in a stull, which the miners had turned
end for end ! Also in some places, notably in coal-mines, the
miners have a habit of removing or tampering with the sta-
tions, thus causing endless trouble unless they are properly
referenced and tested before use.
Stations in shafts or inclines for temporary use may con-
sist of nails driven into the edge of the shaft timbers, or in a
sprag, and are usually set at an angle such that they can be
sighted from above and below. Stations are very often made
of a piece of tin about 3" by 4" with a cross cut in it. This is
covered with tracing-cloth and nailed to a plank over a hole
or notch so that the cross, when illuminated, can be sighted
from both directions.
Stations are illuminated for sighting by holding a light
close behind them. The best method is by holding a piece of
tracing-cloth or oiled paper between the station and the light,
thus presenting a comparatively large illuminated surface on
which the station and cross-wires may be seen. Plummet*
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MINING SURVEYING. 379
lamps have been used ; they are simply ordinary mine-lamps
made in the form of a plummet and swung by a bail so that
the flame hangs true. It is sometimes necessary to illuminate
the cross-wires in taking sights, as well as the stations. It is
generally sufficient to hold a candle near enough in front of
the object-glass to throw light into the telescope-tube ; in
other cases a metal or paper reflector is used, while some
mining transits have a hollow axis through which light may be
directed. The difficulties of properly lighting the stations
and cross-wires vary with the length of the sight, the condi-
tion of the air, and the character of the rock. In coal-mines
it is much more difficult to get sufficient light owing to the
absorption of light by the black walls, etc. ; those coal-mines
that are not fiery use oil-lamps and torches. In metal-mines
candles are much used, except in very wet places, where the
falling water would extinguish them. A good candle, where
it can be used, makes the best light for the surveyor, giving a
clear light, without soot or smoke, and suitable for reading and
sighting. Electric mine-lamps have been devised, but have
not yet come into general use.
2746. To Carry the Meridian into the Win^.— First. By
means of the Transit, — If the mine is entered by adit, tunnel,
or slope, the meridian is carried in by ordinary underground
traverse, to be described later.
If the mine is entered by an incline or a crooked shaft, the
line is run in with a transit provided with an auxiliary tele-
scope. This method is of wide application and is much used
by engineers. It is adapted to that class of mines working
more or less vertical veins, where the shaft or incline follows
the vein on its dip. One shaft surveyed by the authors was
so crooked that in 600 feet eight sights were necessary, yet
the bottom of the shaft was only a few feet away from where
it would have been if the shaft had been vertical. In this case
the bends in the shaft limited the length of the sights and at
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380 SURVEYING.
some points necessitated stagings between the levels. When
the incline is straight, as in the case of some of the largest
mines, the length of the sights is only limited by the length of
the tape, the condition of the atmosphere, etc.
It is easier to run down an incline than up, on account of
the greater ease in measuring the distances. In any event it is
more difficult to see down than up, except in very wet places,
on account of the greater difficulty in lighting the stations.
Great care should be taken in reading the vertical angle, as
a small error in such a case changes appreciably the horizontal
distance between the stations. The accuracy diminishes as the
steepness of the sights increases, because the azimuth is deter-
mined by the horizontal projection of the line of sight, and this
base line becomes very short with steep sights. A steep sight
implies a great difference in elevation between foresight and
backsight ; thus an error in the adjustment of the transit is
greatly magnified, and must be very carefully eliminated in
important cases, by reversal, striding levels, etc. A very small
error in sighting to, or setting over, stations marking the ex-
tremities of a short base line will be material : e.g., suppose the
shaft will allow only a five-foot base line ; then if the sights
were lOO ft. long, the vertical angle would be about 87° 8',
Now, if it is admitted that there is a probability of the com-
bined error of setting and sighting being .003 ft. at right angles
to the plane through the stations, the error would be tan"^[ — ^ j
equal 2 minutes. This is a compensating error, consequently
the square root of the total number of errors probably remains
uncompensated. Thus it is seen that it would take great care
to run, say, 400 ft. down the shaft with an error not more than
3 or 4 minutes. From the above discussion, and also from the
results of experience, it follows that for a vertical shaft the
meridian may be carried down more accurately and more
quickly by plumb-lines than by use of the mining transit. This
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MINING SURVEYING, 381
statement holds with more force with a deep shaft. No defi-
nite rule can be given in regard to this work, as the details in
almost every case are different and much depends on the in-
genuity of the surveyor and assistant.
Second, By use of Plumb-lines,'^ — The plumb-bobs used are
of lead or iron, and weigh from 5 to 20 lbs. according to the
depth of the shaft and the condition of the same. Up to per-
haps 1000 ft. the weight might be from 8 to 12 lbs. with No.
22 copper or soft steel wire ; for shorter distances bobs weigh-
ing from S to 8 lbs., with No. 24 wire, will do ; while for greater
depths or heavier bobs it may be advisable to use No. 20 wire.
Copper wire is generally to be preferred, although it stretches
more. The plumb-bobs should swing in a bucket of oil, mud,
or other liquid that will retard the vibrations. The point of
the bob in swinging traces an ellipse. If we do not choose to
wait for the bobs to stop swinging, a board may be placed
close to each wire and at right angles to the plane through the
two wires and the vibrations read for some time by means of a
scale, then a mark is made opposite the mean position and the
sight taken. Otherwise the surveyor waits until the vibrations
have stopped or have nearly done so, and then takes the sight.
With this method the observations on the wire must be con-
tinued long enough to make sure that the line is stationary and
not merely at some point in its slow vibration. There is in
use in the Pennsylvania coal-mines a bob made of two flat disks
of iron connected by ribs and weighing 20 lbs. ; this is specially
designed so that it may be left until it has assumed a stationary
position.
There are several different conditions met with in plumbing
shafts, which naturally divide the subject for systematic dis-
cussion. These cases will be taken up in order as follows :
First. When there are Two Shafts, — Hang a plumb-line in
each shaft at A and B, Fig. 96, then on the* surface run a
traverse from A to By from which the true azimuth and length
* The operator must guard against constant air currents, however
slight, as these may cause a constant deviation in deep shafts, even with
very heavy bobs. See Engr, and Min, Jour,, Apr. 26, 1902, p>. W^i^
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3^2
SURVEYING.
of the direct course A-B can be calculated. Underground
assume the bearing of the first course (i) to A and run a
MAIN SHAFT
SURFACE SURVEY
UNOERQROUNO SURVEY
Fig. 96.
traverse to the second plumb-line at B and calculate the
azimuth and length of the underground line A-B. The differ-
ence in the azimuths found by the two surveys will give the
correction to be applied to the underground azimuths to bring
them to the true meridian. For a check on the work the hori-
zontal distances resulting from the calculation of the two
traverses should agree.
In Fig. 96 it is assumed that the surface survey determined
the azimuth of A-B as 270*" 10', and that the azimuth as
determined from the underground survey with an assumed
meridian was 275° 20', from which it follows that all the
azimuths of the underground survey should have 5° 10' sub
tracted from them to read from the true meridian. This
method, or a modification of it, should be used whenever
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MINING SURVEYING. 383
possible, on account of the accuracy easily attainable by it.
The base line on which the meridian depends being in this
case the length of the line between the plumb-wires, cor-
responding to A-B in Fig. 96. This method is often used for
running from one level to another in a mine where the main
shaft and a winze or man way can be used.
Second. When One Vertical Shaft is the Only Entrance to
the Mine. — In this case two plumb-lines are hung in the
shaft as far apart as possible. They are placed in this way
because the longer the base line the less a slight discre-
pancy will affect the meridian determined from it. Before
letting down the wires it is usually well to make an exam-
ination of the shaft to determine the most advantageous
way of hanging them; keeping in mind the desirability of
a long base line, but arranging the wires so as to be most
available, at the different levels, for taking off the line. It
often happens that the head-works limit to some extent
the position in which the wires can be hung. This part of
the work requires good judgment and experience on the part
of the engineer.
In the larger mines, with shafts of three or more compart-
ments, it is, as a rule, easy to get a good base line, but it is often
necessary to carry a meridian into shafts much smaller but of
considerable depth, and where the several levels run off at dif-
ferent horizontal angles from the shaft. In some cases the
operation of plumbing will have to be carried out two or more
times in order to connect with the different levels.
After the preliminary examination the wires are let down
the shaft with a small plumb-bob or other weight, the heavy
bobs being attached at the bottom. The transit is set up on
the surface so as to be in line with the position of the wires
determined upon. One wire is then sighted directly, and the
other brought into line.
The wires may be suspended from heavy spikes driven into
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384 SURVEYING,
the timbers or into a cross-beam put in for that purpose, or
any other convenient way that furnishes a solid support. A
very important point is to make sure that the wires hang per-
fectly free. This may be determined by passing a light slowly
around the wire at the bottom and observing the same from
the top. Sometimes it is necessary to climb down the shaft
and inspect the wires in detail, or the support may be given a
measured movement and the wire watched to see if a corre-
sponding movement takes place below. In a well-timbered
shaft using a bucket, a man can be lowered slowly without dis-
turbing the wires, and their position thus fully determined.
In order to get the instrument below, the wires and sup-
ports are so arranged, if possible, that they will allow of the
bucket or cage passing down while the wires are swung tem-
porarily against the timbers. If this cannot be done, and only
one transit is available, it will have to be carried down the
ladders. At the various levels of the shaft, the transit must
be set in line with the two wires at a distance of from
5 to 15 ft. or more from them, according to the exigencies
of the case. This requires great care, especially where cur-
rents of air or falling water prevent the wires from becom-
ing absolutely stationary. The vibration is so slow in many
cases that there is danger of taking the wire in a wrong
position. A plumb-bob swung with a 500-ft. cord would re-
quire over 12 seconds to vibrate in air, and if retarded by
swinging in oil or water it may take several times as long. As
soon as the transit is in place its position should be marked by
a permanent station and another one put ahead, so that at
least two stations are available for orientation. It often hap-
pens that it is not convenient to establish a permanent station
where the instrument sets, after transiting in, but one may be
set ahead and a backsight station fixed at some convenient
point on line.
When this has been done at each level where the azimuth
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MINING SURVEYING. 3^5
is to be taken off, the plumb-lines are removed from the shaft
and the different levels run out at the surveyor's convenience.
If, in order to get a long base line, the wires are placed so
that they are not in line with the levels, it becomes necessary
to set the instrument at one side in the level and tie in to both
wires, thus obtaining the azimuth by the solution of a traverse
or triangle as in the first case, where there are two shafts. This
method is sometimes necessary, but as a general rule it is not
to be recommended, unless a much longer base line can be
had by its use.
The depth of the shaft can be determined by measuring
directly from level to level, or from point to point down the
shaft ; or it may be found by bringing the wire up over a pulley,
under constant tension, and measuring it with a tape as it is
drawn up.
With the short base lines that are necessary in most shafts,
every precaution must be taken to insure good results. With
a ten-foot base line an error of 0.006 ft. or less than one six-
teenth of an inch, if at right angles to the line, would mean an
error in azimuth of 2'. This, in a distance of one mile, would
throw the position of a point about 3 ft. out of place. On the
other hand, a case is mentioned in the Trans. Am. Inst. Min.
Engrs.f where shafts 200 ft. deep, but so twisted as to. give
base lines &' to 18" long, were plumbed and connections made.
The connections were, however, in no case far from the bottom
of the shaft.
Third. When it is necessary to carry a meridian down a
sfiallow but steeply inclined working, and an instrument with an
auxiliary telescope is not available. This can be done by
stretching a wire down the incline and by suspending one or
two plumb-bobs near the top and bottom of the incline, thus
enabling the transit to be lined in both above and below and
the meridian transferred. When it is inconvenient to hang the
plumb-bobs on the wire, they mav be hung tangent to it.
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386 SURVEYING.
"^Fourth, By Alignment of a Wire from the Surface. — If the
bottom of the shaft or incline is not too far away, the method
used in the Severn tunnel may be employed. A wire is
stretched lOO ft. or more into the drift at the bottom, the
ends passing over screws with which the wire may be accu-
rately aligned by the transit, for the few feet visible from the
top of the shaft. This is quite an accurate method, where
applicable, if a heavy transit with a large telescope is used.
The work of carrying the meridian into the mine should,
in very important cases, be gone through with, independently,
at least twice, besides using every precaution and check in
measuring the angles and distances in each case.
This part of the survey is generally the most expensive for
the mine, as it stops work in the shaft. It also generally pre-
sents more difficulties and requires more judgment* on the
part of the engineer than any other part of the survey.
274c. Underground Traversing. — This is ordinarily car-
ried on much the same as on the surface, with the exception
of a number of details due to darkness and other conditions
peculiar to underground work. The measurements are usually
made on the horizontal in running out an ordinary level or
entry. If for any reason the elevations of the stations are re-
quired with a fair degree of accuracy, it will be necessary to
read the vertical angles and the height of instrument at each
station. For very exact work, as in running out grades, a
leveling-rod should be used.
The H.I. is generally recorded positive when the instru-
ment is above the station, and negative when it is below.
The necessity for taking the H.I.'s can be avoided by measur-
ing the distances and vertical angles both ways from the alter-
nate stations occupied. This method carries the elevation
forward and gives the elevation of stations measured to.
The three-tripod method of traversing underground makes
use of two target-lamps mounted on an axis similar to the one
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MINING SURV^YINC, 387
on the transit, so that the cross of the target is the same dis-
tance above the leveling head as the cross-wires of the transit.
This involves the use of the short-axis transit, so that it can be
lifted ofl above the leveling head and be exchanged with the
lamp-target.
The instrument being set up over any station, one tripod
with lamp is set up over the backsight station, and the other
over the forward station. The sights are now taken to the
target-lamps, and the measurement made from the horizontal
axis of the instrument to the centre of the target. The
vertical angle is read and the H.I. measured. Now the
instrument-head is lifted out of its socket and carried to the
forward station, replacing the target-lamp at that point, the
target-lamp being taken back to the former instrument
station. The rear target and tripod is next moved up to the
forward station, and the work proceeds. The three-tripod
method, it is claimed, is expeditious and accurate ; its dis-
advantages are the extra cost of apparatus and the difficulty
of carrying so much around. It would seem to be adapted to
the needs of a surveyor who had a large mine to look after,
and where all the workings were horizontal or not steep
enough to require ladders. These conditions are more often
present in coal-mines than in metal-mines.
On p. 388 is shown a good arrangement for the notes of an
underground traverse and connection to the surface. Traverses
are usually run through all the main workings of the mine.
Starting at each level from the meridian previously brought
in, as many permanent stations and bench-marks are established
as may seem desirable. The position of these stations shculd
be noted, and also any irregularities in the drift to enable it to
be platted correctly. For very careful work, offsets from the
traverse lines are taken at short intervals. In some cases it is
necessary to set reference points near the stations so that these
can be replaced if destroyed.
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388
SURVEYING,
(Left-hand page.)
UNDERGROUND SURVEY.
March i6, 1899.
(Right-hand page.)
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Where winzes or manways extend from one level to an-
other, it is a good plan, when convenient, to run connections
through them, as a check on the work if for no other purpose.
The work when completed gives data for a general map of the
mine, but supplies no detailed information in regard to the
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MINING SURVEYING. 3^9
stopes. When the latter is necessary a line can be run into
the stopes from the most convenient reference station.
Stope-work consists generally in determining approximately
the amount of ground broken and the extent of the workings ;
therefore if one or two points in the stope have been connected
with a permanent station the rest of the measurements may
often be taken with the tape without reading angles.
For such surveying work or for any approximate work,
especially in confined spaces, the German swinging compass
and clinometer is very good. It is hung on a cord or wire
stretched from one station to another. The cord corresponds
to the line of sight between stations, as its direction and angle
of elevation or depression is read directly by compass and
clinometer. One of the chief advantages of this method is
the case with which coordinates can be measured directly
from the cord, at any point, to the walls or boundaries of
the confined space. The hand compass and clinometer is
much used in measuring up stopes, and is often sufficiently
accurate for use in the examination of mine- and ore-bodies by
an engineer making a preliminary report.
274<f. Underground Leveling. — For underground leveling
the ordinary wye level is used. It should be provided with a
heavy adjustable tripod in addition to the ordinary one. In
case a level is not available the transit can be used unless
extreme accuracy is wanted. A convenient rod is one 5 ft.
long and capable of being extended to 9 ft. The rod can be
read by target directly when visible, otherwise a small steel
rod like a knitting-needle may be soldered on to the target at
the zero line so as to project two or three inches, then a paper
and light held behind it properly will enable the target to be
set. Considerable care should be used in underground leveling
where connections are to be made by tunnels or drifts, as for
good haulage and drainage it is necessary that the grades
connect well. /
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39<^ SURVEYING,
274e. Mapping the Survey. — If the mine workings lie in
one bed, horizontal or nearly so, as in many coal-mines, one
plan of the workings may be sufficient. In ordinary metal-
mines, where the workings extend more or less vertically
downward, the survey is properly shown by at least three
maps: first, the plan or projection on a horizontal plane;
second, the longitudinal section, generally the projection on a
vertical plane coinciding as near as may be with the general
direction of the levels ; third, the transverse section or projec-
tion on a vertical plane at right angles to the longitudinal one.
(See Plate V.) In addition to these three maps it may be
desirable to make true sections at special points, and some-
times a separate map of the surface survey is made, although
this is usually combined with the plan. The maps should have
a title giving the name of the mine and location by mining
district, county, and State ; also there should appear ^he name
of the surveyor, date of survey, meridian used, and scale.
The map should be on cloth-backed paper or tracing- cloth,
and may show on the plan the position, number, and elevation
of the permanent stations with the bearing and length of the
lines joining them, or the coordinates of the stations, accord-
ing to the system used. The advisability of showing these
data on the map depends on the use that is to be made of
them and must be decided according to the nature of the
case.
It is often a good plan when the different levels of the
mine are in nearly the same vertical plane to plat them in with
inks of different colors, in order to prevent confusion. (See
Plate V.) There is an objection to this when blue prints are
to be made, that the colored inks do not print well. The
number and arrangement of the maps vary greatly among the
different surveyors even when the same conditions are to be
fulfilled. A very convenient way of making the maps is to have
a main map on heavy paper on which all the workings are
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MIKING SUKVEYING.
391
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platted and kept up to date; a trac-
ing is made, and kept up with the main
map, from which blue prints can be
taken as desired. The scale used va-
ries from 10 to 200 feet per inch, de-
pending on the extent of the workings
and the use to which the maps are to
be put. A very common scale for
working maps of the metal-mines in
Colorado is 20 feet to i inch. Such a
map should be accurately drawn, the
angles being laid off by means of a large
metal protractor or by latitudes and
departures, natural tangents, chords,
etc. It can then be used as an exact
check on all work and calculations ; the
distances scaled off will be close enough
for running air connections and for use
in problems of similar importance.
Where the workings are very ex-
tensive the use of coordinates in plat-
ting and in keeping the records may be
advisable. In such a case the simplic-
ity and regularity of the method gives
it a great advantage. It also allows
of using the formulae of analytic geom-
etry in the solution of problems.
The coordinate method is used in
the big mines of South Africa not only
on account of the advantages men-
tioned, but also because the maps that
must be filed with the officials at stated
intervals are required to be platted in
this way, each point being given by its
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392
SURVEYING,
X and y coordinates in metres. The reduction of the notes
should be in a book kept for that purpose and may be
arranged somewhat as shown on p. 391.
To explain the method of constructing the maps, let A-B^
Fig. 97, be any course whose length, azimuth, and angle of
depression are known. For the
plan multiply the length A-B by
the cosine of the vertical angle
which projects it into the length
0-B in the figure. Suppose the
plane of the longitudinal section
has the direction O-L^ making
a known angle with the meridian.
Change the bearing of the course
so as to refer to 0-L as a meri-
dian. Now the latitude of A-Ly
the course as it appears in the
longitudinal section, will be A-O^
the difference in elevation be-
tween A and B^ and the departure O-L^ the projection of 0-B
upon the assumed plane. In the same way it is evident that
A-T is the course as it appears in the transverse section,
and that it is given by its projections A-0, the difference in
elevation of the two stations, and 0-T^ the projection of 0-B
upon the assumed plane. The student could readily compute
these values from the given data.
274/. The Problems of Underground Surveying. — First,
To make a connection, i.e., to find the bearing of a line joining
two given points and the horizontal distance and difference
in elevation between them. This is one of the most important
problems, as it includes all cases where connections are to be
made from one part of the mine to another by sinking shafts
or driving cross-cuts, winzes, etc., along the course figured,
and it requires more accuracy than is necessary for mapping
Fig. 97.
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MINmC SURVEYING. 393
or similar purposes. This problem can always be solved
directly whenever it is possible to run a traverse from one
given point to the other, no matter how devious or round
about the route taken. For on reducing the traverse lines and
calculating the lost line the bearing and horizontal projection
of the required line are known, while the difference in eleva-
tion of the two known points gfives the other projection, and
consequently the grade between the points. If this principle
is understood, the surveyor can turn attention to the real dif-
Acuities — the liability to error brought about by the necessity
for short and steep sights, the interference of water, bad air,
steam, the lack of light, and the cramped places. The young
surveyor is advised in such a case to pick out the most un-
reliable sights, assume a probable error, and figure out what
difference it would make in the connection. If not an allow-
able error, the survey should be gone over with more care if
possible, and in any event getting average values that should
be nearer the truth.
Too much stress cannot be laid on the importance of the
care to be exercised in running connections, as there is nothing
the mining surveyor's reputation depends on more directly
than his uniform success in this matter. In fact, a failure in
such a case may involve a large loss to his employer, or if he
has guaranteed the work the cost comes out of his own pocket.
On the other hand an error in many cases cannot be remedied,
but results in a permanent injury to the mine.
Example : Showing the Method of Making an Underground
Connection. — It is required to give points in cross-cut on second
level to start a vertical upraise to connect with shaft at first
level. The method is as follows : Set up in the cross-cut on
the first level and carefully tie in two corners of the shaft and
run to station (2) in the level, thence to station (3) at the top
of the winze, then down the winze and along the second level
to the cross-cut, and thence to the breast of cross-cut. Per-
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394
SURVEYING.
^
PLAN
WMy^^^^^^^^^mm^^mmi-
LONGfTUOmAL
SECTION
^t
■m
manent points are put in here : station (7) near the breast, and
station (6) further back. These points will serve to orient by,
for the latter work. Plat the
data thus obtained and find how
much further the crosscut should
be run and where the station is
to be cut. Calculate a traverse
from one corner of the shaft
to station (7), and the bearing
and length of one side of the
shaft. When the station has
been cut out set a point for one
corner of the shaft from station
(7); as figured then from this
corner the points for the other
corners of the shaft may be set.
A satisfactory way of establish-
ing these corners is to set wooden
p^^l^^,,^ plugs in holes in the rock below,
^ ''^ with nails to mark the exact
points. Plumb-lines can be sus-
pended over these nails and the
upraise thus kept in line. It
should be distinctly stated wheth-
er the points given are inside or
outside the timbers. Generally it
is best to put them inside the
lers cut out as much more as neces-
J L
P
CROSS COT W.
ij
TRANSVERSE
8ECTI 0 N ./ym^//^/m
^ \ CROSS COT ^.
^ ] CROSS CUT l^
Fig. 98.
timbers and let the m
sary.
The difference in elevation between the top of the station at
the end of the cross-cut and the bottom of the shaft will be
the length of the upraise. Great care must be taken in this
work if the shaft is to be timbered up. Timbering both ways
in a shaft is to be avoided if possible, because an error of two
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MINING SURVEYING.
395
IBs^
or three inches at the point of meeting is bad, whereas if not
timbered until connection is made, that error would not be
material on account of the leeway that must be allowed for the
timbers.
In case it is required to run a drift in a certain direction for
a connection, two plugs on the hne will be a good guide for
the miners. If plumb-lines are hung on these plugs, the line
may be followed fairly well.
Second. To establish property lines underground. This
problem is probably of more frequent occurrence than any
other, and is also very important. In many cases the boundary
line passes through a rich ore-shoot, and the necessity for care
in establishing it may be readily
seen. It is not uncommon for
such a line to be determined
underground by a joint survey
conducted by the surveyors of
the adjoining properties. The
following example will serve
to illustrate the method of
solution.
It is required to establish un-
derground the line joining Cor.
No. I with Cor. No. 2. This
may be done as follows : Deter-
mine the course of the line and
run from the corner or some
convenient point on the line to the shaft, thence down to the
level by the most convenient method, in this example by
plumb-lines. From the station at the bottom of the shaft run
out the drift to station 105, and from this station take the
bearing of the drift toward the line. Now calculate from the
traverse a direct course from Cor. No. i to station 105. Having
calculated the length and bearing of this line, and observed the
LONQITUDINAL SECTION
•^ ;;,y>:<r;^;;,^////^
iw^
'-z^^m^M
Fig. 99.
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3g6 ^VRV EYING.
bearing of the boundary line and of the drift, solve a triangle
which will give the distance from station 105 to the boundary'
line. If it is desired to have the exact point marked, a plug
may be put in on line. Sometimes the line cuts diagonally
across the drift, or in any case where there is room enough it
may be desirable to set up over the station on line and place
two plugs or marks on the boundary line, one in each wall.
Third. To find the bearing and length of a line to intersect
a vein of known dip and strike. This includes problems in-
volving the consideration of the vein as a plane with a known
dip and strike. This class of problems can generally be solved
only for approximate results, because it is very rarely that a
vein is a true plane, as, even when not faulted, the variations
from a constant dip and strike may be considerable. It can
only be said in such cases that if the vein continues on a con-
stant strike and dip it will be intersected at a certain point.
Although this uncertainty exists, it is nevertheless most im-
portant to know at what point to expect the vein.
As an example, let it be required to find the distance that
the tunnel (i) to (2), Fig. 100, must be run to intersect the
vein (3)-{4)-^^, there being an incline at (3) and an outcrop of
the vein.
Set up at station (i) and sight to (2) ; this determines the
bearing required. Run a traverse to station (3) at the incline
and find the dip and strike * of the vein, then calculate the
closing line from (i) to (3) and solve a triangle for the horizon-
tal distance from (3) to (4), this latter being the point where a
vertical plane through the tunnel cuts the outcrop or strike of
the vein, on the surface, and from (i) to (4). If the point (4)
is found on the ground and the difference in elevation between
it and station (i) determined, the required length is calculated
* The dip is the steepest inclination of the plane of the vein to the
horizon, and the strike is the horizontal direction of its outcrop upon the
surface.
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MINING SURVEYING.
397
as follows : Assume the tunnel is horizontal, then the distances
a to (4) and a to (i) are known, (4)-^ being a vertical line.
A plane passed through ^-(4) and perpendicular to the
VERTICAL SECTION ON LINE OF TUNNEL
Fig. too.
plane of the vein intersects it in the line ^-{4), and intersects
the horizontal plane through the tunnel in a-b. Having
^-(4), solve a right triangle for a-b, the acute angle being
90° — the dip ; then solve the right triangle abc for ac^ the
acute angle >^, being the angle between directions of strike and
tunnel ; then ac -f- ^(2) is the distance the tunnel is to be
driven. If it is driven to grade, it can easily be figured what
difference in length would result. When it is required to drive
a tunnel at right angles to the strike of the vein, i.e., to run
the shortest tunnel from a given point to intersect a given
vein, the above solution becomes much simpler.
274^. Surface Surveys. — It is always necessary, or at least
desirable, to verify and tie in the monuments that mark the
boundaries of the mining property, and to establish the true
meridian very carefully, at a point convenient for connection
with the underground surveys. If expensive plants are to be
erected, it may be advisable to make a topographical survey
with transit and stadia for the purpose of locating shafts,
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39^ SURVEYING,
buildings, dumps, streams, etc., together with the contour lines
and any important geological features.
Sometimes the traverses of underground surveys must be
duplicated on the surface to find the exact relation of under-
ground work to the surface. This is in case a topographical
survey has not been made, or when exactness is required.
274^. Court Maps. — This name is one given to maps that
are to be used as evidence in court in a mining lawsuit.
When the maps are used in this way the surveyor is generally
called upon to verify the map in court. Court maps may be
classified according to issues involved.
First, When the case is to determine the ownership of
ground, as in an adverse suit, the map usually shows the sur-
face ground of the claim in dispute, the position of workings,
and any point that may have a bearing on the case, but ordi-
narily there is no need of showing the underground workings.
These maps should be accurately made and drawn to a fairly
large scale, and the desired special points should be distinctly
brought out so as to be clearly understood by a jury.
Second. When the suit is to determine the ownership of
veins, etc., as in the endless variety of apex suits of which the
mining laws of the United States are so fruitful a cause. In
this case it is necessary to show the boundary lines and all the
underground workings that have any bearing on the question
at issue. It is often necessary to construct a number of cross-
sections to show the continuity of the vein or the reverse. In
this class of work, especially in important cases, large sums of
money are spent on the surveying and mapping of the prop-
erties. No exact rule can be laid down for this kind of work,
as the conditions vary so widely, but the surveyor should in all
cases study thoroughly the problem in hand so that the maps
may be made to show clearly the matters in dispute.
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MINING SURVEYING, 399
DEFINITION OF MINING TERMS.
Adit, A horizontal drift or passage underground opening from the sur-
face. ^
Apex, The top of a vein, usually wliere it outcrops on the surface.
Cross-cut, A passage or tunnel driven across the course of the vein.
Dip, The angle the plane of the vein makes with the horizontal.
Drift, An underground passage driven along the vein.
Incline, Applied to a passage having a more or less fixed inclination
from the horizontal ; a slope.
Level, One of a number of horizontal passages along, or parallel to, the
deposit, and placed at more or less fixed intervals, generally loo ft.,
for the systematic working of the mine.
Manhole, A small passage from a level into slopes or to the next level
above.
Mill' hole, A passage left from stops to level for dropping down ore or
rock.
Outcrop, That portion of the vein intersecting the surface.
Shaft, A hole sunk more or less vertically downward.
Slope. The workin*^s above or below the levels where the mass of the
ore-body is broken.
Stride, The direction taken by the intersection of the vein with a hori-
zontal plane.
Stulls, Cross-timbers between the walls of the excavation.
Sump, An opening at bottom of shaft or at any level for the collection
of water.
Tunnel, A horizontal passage from the surface ; properly speaking it
should be open at both ends, but it is not always so used in mining.
IVinse. A shaft sunk from a level,
upraise, A shaft excavated upwards from below.
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CHAPTER XII.
CITY SURVEYING.*
275. Land-surveying Methods inadequate in City Work.
^— The methods described in the chapter on Land-surveying
are inadequate to the needs of the city surveyor. The value
of the land involved in errors of work, with such a limit of er-
ror as was there found practicable (see art. 180), is so great as
to justify an effort to reduce this limit. Comparing the value
of a given area of the most valuable land in large cities with
the value of a like area of the least valuable land which a sur-
veyor is ever called upon to measure, the ratio is more than a
million to one.
This view is emphasized by the manner of use. On farm
lands the most valuable improvements are placed far within
the boundary-lines, but the owner of the city lot is compelled
by his straitened conditions to place the most costly part of
his improvements on the limit-line. His neighbor's wall abuts
against his own. The surveyor, who should retrace this line
and make but a small difference of location, would get his
clients and himself into trouble. Both the value of tlie land
and the manner of its use demand increased care. The modi-
fications of the methods used in land-surveying to meet the
requirements of work in the city will be treated in this chapter.
Much of the work described furnishes data for the solution of
engineering problems, but the obtaining of the facts falls en-
tirely within the definition of surveyor's work.
* This chapter written by Wm. Bouton, C.E., City Surveyor, St. Louis, Mo.
See also Appendix G.
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CITY SURVEYING. A^^
276. The Transit is used exclusively, but the common pat-
tern may be very materially modified with obvious advantage.
Seeing that the magnetic needle* is never precise and seldom
correct, it should be wholly discarded in the construction of
the city surveyor's transit. The verniers can then be placed
under the eye, the bubbles can be removed from the standards
and placed upon the plate of the alidade, and the standards
themselves can be more firmly braced. By these changes a
steadier and more convenient instrument is secured, when the
useless and somewhat costly appendage of a needle-box is out
of the way. The adjustable tripod head and the levelling
attachment are always convenient. For topographical work,
the vertical circle, or a sector, and stadia wires are essential,
otherwise the methods used must be primitive. The ther-
mometer which is needed in order to make the proper correc-
tions for temperature may be conveniently attached to one of the
standards facing the eye-piece of the telescope. The danger
of breaking the tube while handling the instrument may "be
obviated by a guard sufficiently deep to protect the bulb, made
open on the side toward the observer.
277- The Steel Tape is generally used for measuring. The
legal maxim that "distances govern courses,*' when interpreted,
means that, using customary methods, the intersection of two
arcs of circles, centres and radii being known, is a more definite lo-
cation of a point than the intersection of two straight lines whose
origin and direction are likewise known. The fact is, the inter-
sections are not more definite. The maxim grew into authority
when the compass was pitted against the chain. With the
transit to define directions of courses, and the chain still to
measure the distances, such a maxim would not have voiced
the results of experience, but would have been sheer nonsense.
* The needle finds its proper place where checks are not so abundant, and in
classes of work in which a close and rapid approximation is of more value than
orecision.
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402 SURVEYING,
The ordinary chain has too many gaping Hnks, and the brazed
chain too many wearing surfaces, to be kept in very close ad-
justment to standard length. Its weight is such as to make the
** normal tension" (see p. 392) impracticable ; hence the effect
of slight variations of pull is much more marked than if the
tape is used. Graduated wooden rods were used until i860 to
1870. They were unwieldy when twenty feet long, and were
still so short that the uncompensated part of their compen-
sating errors was a matter of considerable moment. Every
time the pin is stuck or a mark made at the forward end of the
tape or rod, the work is a matter of skill and involves an error
dependent on the degree of skill attained. When the measure
is brought forward, its proper adjustment in the new position
is a matter requiring skill. These errors are compensating, but
the resultant is not zero. The use of the plumb-line is another
source of compensating errors which are reduced by an increase
of length in the measure. First, the number of applications
varies inversely as the length of the measure ; second, using the
rod, it was necessary to work to the bottom of ravines and gul-
lies and then work up again ; now the long tape spans them at
a single application. The minus errors due to imperfect align-
ment and inaccurate levelling of the two ends have a greater
percentage of effect when the measure is short than when it is
long. The longer tape brings with it some other sources of
error. When used suspended at the ends there is a minus
error on account of the sag of the intermediate parts, and a
plus error from elongation due to tension ; there is also expan-
sion by heat, which produces an error which may be plus or
minus as the temperature at the time and place is above or
below that for which the tape is tested. The effect of sag
increases very nearly as the cube of the length when the ten-
sion is constant. When, to counteract this increase, the
pull is made greater than a man can apply uniformly under
all conditions — at his feet or above his head — there come
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CITY SURVEYING. 403
irregularities from this cause. The limit of length of tape
which it is practicable to use will be determined by the condi-
tions of the work, and should be such that the increase of
length involves greater error than it eliminates. On account
of convenience in keeping tally, 5ofoot and 100-foot lengths
are generally used. In a level country the 100-foot tape is
preferred.
There are tapes made with the purpose to eliminate the-
errors which arise from the free-hand pull, the inclination of
the tape, and the temperature. They carry a spring balance,
a bubble adjusted to the desired pull, a thermometer, and a
means of adjusting the length to the given pull and tempera-
ture. The effort is laudable ; but, probably on account of the
number and form of the wearing surfaces, they have not yet
met with general favor. Further progress may be made in this
direction.
LAYING OUT A TOWN SITE.*
278. Provision for Growth.— Cities grow. It is very rare
that the considerations which should have governed have been
given any place in determining upon the plan of the original
town. The considerations first in importance are topographi-
cal. What are the natural lines along which business will tend
to distribute itself? To what form of subdivision can it adapt
itself with the least resistance? Where is the best harbor,
the lake or river front, or the railway line? Ordinarily the
land immediately adjoining such natural features is not best
used when used as a street, but when occupied by private
* For principles governing the laying out of new cities, see a valuable paper
entitled Practical and ^Esthetic Principles for the Laying Out of Cities, by J.
SttSbben, Commissioner of Public Buildings and Assistant Burgomaster, Cologne,
Germany ; read before the World's Engineering Congress at Chicago, 1893, and
• published in the Trans, Am, Soc, Civ, Eng,^ vol. xxix., p. 632.
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404 SURVEYING,
docks, or along a railway by warehouses and factories having
switching facilities without crossing public streets. The
streets parallel to such lines should be of ample width,
easy grade, and continuous but not necessarily straight align-
ment. Much of the heavy hauling will be along such streets.
In the business part of the town the cross-streets should
be so frequent as to make the blocks approximately square.
In the residence portion alternate streets in one direction may
with advantage be omitted: this saves the expense of unneces-
sary streets, and permanently lightens the burden of taxation.
Which fronts are on all accounts most desirable in the par-
ticular locality will determine in which direction the blocks
should be longest.
279. Contour Maps. — Another phase of topography de-
mands attention. The sites of suburban towns may generally
be best handled by laying out streets and lot lines in conformity
to the undulations of the ground. Additions to the city may
also have characteristic features that can be preserved with
advantage. For all such cases a contour map is very useful
to one who is able to interpret it. The making of all the
ground available, and sightly points accessible, and at the same
time so locating the streets as to secure economical grades, — in
short, the judicious handling of the whole subject is facili-
tated by the study of the contour map.
280. The Use of Angular Measurements in Subdivi-
sions.— Shall subdivision lines be located by an angle with the
^ street on which the lots front or by
eY 400 if distances from the next cross- st ree t ?
Must distances govern courses, what-
ever methods are used ? Let us sup-
1^, pose, for illustration, that it is re-
* quired to locate lot g in the accom-
FiG. 102. .
panying sketch (Fig. 102). Suppose,
farther, that it is possible to measure each of the linQ^ qh^
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CITY SURVEYING, 4^5
and dc with a maximum error of i in 5000 and that the
conditions are such as to produce opposite errors in the
two lines. Then, ist, the resulting error in locating the line
be, i.e. {ab — dc) will be y^ X 400 X 2 = o. 16 feet. The
sine of the angle by which the angle A' differs from A will be
^1^=: .00107. Hence the change of direction on account of
the errors in measurement is 3f minutes. 2d, the line ej
must be distant from ^^3f X 150 feet = 550 feet, in order
that, under like conditions, if it is measured instead of dc, the
change in direction shall not exceed one minute. Or the loca-
tion may be made by measuring the line ab, or a line near to
it where favorable conditions exist, and then repeating ba
the same man being fore-chainman ; the principle of reversal
is thus applied to this measurement. Then measuring A ^=^ A
and repeating the angle, reading both verniers, the error is
brought within the maximum error in the pointing power of
the instrument. In order to locate be from ab parallel to ad^
two monuments marking the line ab need to be known. The
other method requires also a monument locating the line ae.
It thus appears that when the side-lines of lots are located
perpendicular, or at any other known angle with the street
upon which the lot fronts, it is susceptible of more accurate
location than by two (front and rear) measurements, unless the
usual limit of error can be greatly reduced. While it is not
likely that maximum errors of opposite character will fall to-
gether affecting the work on the same lot, it is quite as im-
probable that the maximum error in measuring an angle
should vitiate the work of the transit. It is probably quite as
easy to reduce the maximum error in measuring an angle to
half a minute as it is to keep the maximum error in measur-
ing distances down to i in 10,000.
281. Lasring out the Ground.* — The work of putting the
plan upon the ground is a very important one. This is about
* See Appendix H, on City and Village Plats in Michigan.
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406 SURVEYING,
the worst possible place to do hurried and inaccurate work
Fences or other styles of marking possession which limit the
contour map cannot be relied upon as defining the property-
lines. These lines must be accurately located in relation to
the streets of the town or of the addition, in order to make
practicable such exchanges or sales as may be necessary to ad-
just property-lines to the new block-lines. This method is
preferable to that which adjusts block-lines to the original
property-lines.*
As a framework for the whole survey an outline figure,
generally a quadrilateral, of sufficient dimensions, and so
placed that it can be permanently marked with monuments
which will remain accessible when the town is built up, should
be located with especial care. All lines should be measured,
all angles observed, and all practicable checks introduced.
This figure must close absolutely ; that is, the record of the
work when completed should be mathematically consistent.
Unreasonable errors are to be eliminated by retracing the work.
In the adjustment which distributes the remaining errors each
part of the work should be weighted (art. 174, Rule 2), for it
is very rare that a land-survey is completed under such con-
ditions that the man who does the work would be justified,
while these conditions are fresh in his mind, in assuming that
the probability of error is alike at all points. The angles ad-
mit of adjustment independently of the length of the lines.
That distribution of the angular errors which reduces the errors
of measurement to a minimum has such weight that it can
be overruled only by the most positive evidence that the cor-
* In some places this idea of the private interest of the proprietor, some-
times private spite, is carried to such an extent that it would seem that each
man's farm or /^rden patch was especially fitted to be a town by itself, laid
out with utter disregard to the towns which others are in like manner laying
out upon adjacent farms. In this practice the interests of the public for all
time are neglected in order to secure a doubtful advantage for one. Where the
custom prevails it is better honored in the breach than in the observance.
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CITY SURVEYING 40;
rections so indicated cannot be the true ones. The distances
are then adjusted to the angles so determined. The re-
mainder of the work of the subdivision is checked upon the
adjusted outline, reasonable errors being distributed and un-
reasonable ones retraced.
282. The Plat to be geometrically consistent— The
necessity that the recorded plat should be consistent lies in
the use that is to be made of it. A parcel of ground de-
scribed by reference to the plat of record should have but one
location, not any one of two or more possible locations, as is
the case when the plat contains errors on its face. In the
course of years the lines of such parcels will be retraced proba-
bly many times, at one time by one method, at another time
by another equally in accord with the plat. If the plat is not
consistent with itself and with the monuments upon the
ground, this error will be pretty sure to find its way into the
lot location. When the fault is with the plat, it matters not
how the monuments are placed upon the ground ; they cannot
mark the chief points and all agree in such a way that if any
two remain and the others are lost the relocation will in every
case be the same. But this is just what the plat is for — to
make a public record of the relation of each part of the sub-
division to every other.
283. Monuments.* — How many monuments shall be lo-
cated, and where shall they be placed ? What material shall
be used and how set? Answering the first question, it is plain
that no more work should be attempted than can be done well.
Better one point and an azimuth than points everywhere and
no two agreeing either in distance or direction with the rela-
tion described by the plat. But so much should be done well
that the labor of locating any point in the subdivision from
existing monuments shall not be excessive. The points
chosen for placing monuments should be such as will continue
to be accessible and will not be ambiguous. The centre lines
* See also Arts. 159, lOo, 161, and 194 in chapter on Land Surveying.
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4^8 SURVEYING.
of intersecting streets are sometimes marked, giving one monu-
ment to each intersection ; others choose the side-lines, giving
four monuments to each intersection of streets. If the blocks
are so long that intermediate points are desirable, points on
the ridges should be selected.
Stone is more often chosen than any other material ; iron
bars, gun-barrels, gas-pipe, etc., are sometimes used, driven
with a sledge ; cedar posts, say 4^^ X 4^ are quite durable, and
hard-burned pottery is sometimes used. Whatever material
is chosen, the foundation, which should be flat — not pointed —
must reach below frost; and the centre of gravity is kept as low
as possible, so that there shall be no tendency to settle out of
place when the ground is soft in the spring. When the tops
are much above the surface of the ground, there is a liability
that they may be displaced by traffic. Probably the surveyor
does not see any traffic, or the prospect of it, when he is doing
his work, but the traffic must come before the work of the
monument can be spared. It is better to bury the stone wholly
and indicate where to dig for it by bearings than to run the
risk of losing the whole work through indiscretion in placing
the monument that marks it. In situations where every rain
storm produces a slight fill it is safe to place the top consider-
ably higher than would otherwise be reasonable. The stones
to be set are so placed in the excavation, with the heavy end
down, that when the top is in the proper position and before any
earth is refilled there is no tendency to fall in any direction ; then
while the earth is being refilled and thoroughly tamped about
the stone, the top is kept in place. It is better that the mark
denoting the point for which the stone stands should be cut
upon before it is placed in the ground. When this is done, if
the mark is worn off by traffic or knocked off by accident, the
centre of that portion of the stone which remains is a very
close approximation to the original point. A slovenly way of
slighting this work is to tumble the stone into the excavation,
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CITY SURVEYING, 409
fill around it pretty much as it happens, push it to one side or
another so that the point will come somewhere on the top, and
then cut the mark wherever the point comes. Stones set in this
way are liable to settle out of place after the first heavy rain,
while frost and rain keep up their work till the stone lies flat
upon its side. If by chance it should keep its place pretty
well and the mark becomes defaced, it might as well be any
loose bit of rock as a set stone, for its centre gives no idea of
where the mark was placed. No one should be trusted to set
corner-stones unwatched who is not familiar with the work
and thoroughly reliable.
Points are preserved temporarily by wooden stakes driven
flush with the ground. The point, preserved by offsets while
the stake is being driven, is marked by a nail. Witness-stakes
driven alongside and standing above grass and weeds assist in
finding the stakes when wanted. Made of half-decayed soft
wood, e.g., old fence-boards, such stakes will hardly last a
sea.«ion ; while durable wood, well seasoned, will last much
longer than any driven stake can be relied upon, since it does
not go below frost, and is liable to be pushed by a passing
wheel or be otherwise disturbed when the ground is soft.
284. Surveys for Subdivision.^ — The purpose of making a
survey before recording a plat of a subdivision is twofold, —
first, to get the information which it is desirable to record;
second, to leave such monuments as will make it easy to locate
any portion when de.sired. The recorded plat should show
sufficient facts to determine the relations of every part to the
whole, and these relations should be shown by methods which
involve the minimum of error, i.e., giving a location which may
be retraced with least possible doubt. The current practice
falls short of this standard at some points which are worthy of
note.
{a) Surveyors seem to have no doubt of the ability of their
field-hands to measure a line, but very seriously doubt their
* See also Appendices G and H.
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410 SURVEYING,
own ability to measure an angle. Angles are measured dur-
ing the progress of the work and are used for determining the
lengths of lines ; these lengths are then made a part of the
record, while the angles which determined them are omitted.
Apparently some things which are dependent have become
more certain and fixed than that upon which they depend. A
proper record of angles would show what lines are straight and
where deflections are made. Deflections which are sufficient
to very seriously affect the position of a brick wall do not show
on the scale of the recorded plat. For example, an addition to
a town extends from Fifth Street to Twelfth Street ; extreme
points are well established, but intermediate monuments are
missing ; and it is required to establish at Eighth Street the
line of a street which a ruler applied to the recorded plat sug-
gests is a straight line. Custom approves that in such a case
the surveyor should try a straight line, there being a mild pre-
sumption in its favor ; but if his straight line agrees with one
wall and disagrees with two walls and a fence, he had better
look further before he comes to a decision. No such doubt
could have existed if the recorded plat had been properly made.
{b) Very few recorded plats show what stones have been
set by the surveyor, or indeed indicate that he has any knowl-
edge that such monuments may ever be useful. If the custom
were once established of noting upon the record the location
and description of monuments, any monument found during a
resurvey, but not shown on the record, would be discredited.
As matters now stand it must be proved incorrect to be dis-
credited— a thing not always easy, for a system of quadrilat-
eral blocks whose angles are not recorded and whose street
lines are not necessarily straight is not theoretically very rigid.
(c) Many plats require measurements to be made along
lines which are easily measured while the land is vacant, but
which will become inaccessible as soon as the property is built
up. The obstacles to be overcome before the result can be
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CITY SURVEYING,
illl
reached by the method described on the record will each add
to the doubt of the accuracy of that result. There are many
ways in which plats are made, which are all justly subject to
this criticism. Two examples will suffice. Irregularly shaped
blocks are sometimes treated as in the annexed sketch, Fig.
103. The outline is subdivided mechanically, and proportional
distances are given on interior lines which are not consistent
with any trigonometrical relation of the exterior lines, much
less with that which does exist but is not recorded. The point
X has nine distinct locations directly from the plat. On the
theory that ab and cd 3ire straight lines, their intersection gives
one ; ad straight, the distances ax and dx give each one ; cd
straight, the distances ex and dx give two. Combine the dis-
tances ax and ex, bx and cx^ etc., and get four more. But this
is not all, for the point x stands related to each of the ten other
points along the line ab, and each of these has also nine loca-
tions which accord with the plat, and our point x may be lo-
cated from either of them or any combination of them when
they have been located by any of the methods described.
Besides the difficulty of determining how interior points
should be located, this fan-like subdivision wastes ground in
each lot which results in wedge-shaped remnants about the build-
ings, which remnants would be valuable if thrown together into
the corners, thus keeping the remaining lots rectangular at the
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412
SURVEYING,
front. The attempt to reach a rectangular front sometimes
fails through inattention to very simple matters, as in Fig. 104.
Here no angles are recorded. The rear corners of the lots are
located along the line ab by distances from ^ or ^; but the
record-depths do not fall upon a straight line. The line ab
should bisect the angle between the block-lines or be parallel
to such bisection in order that with a constant distance along
ab common to the series of lots on each side of that line their
^,,^
<A
*\
* \
\
_\
rviS
^^
-1^\ » \
\ \
r^
"
&
V
^^""'^'^^^
^ » \ J
r-^
•— ^»
% %
^-""""^"V 200'"
17
100
— 5^
16
ft
14
M
It
12
11
100 1
Fig. 104.
angles with their respective fronts may remain constant. In
the case given every lot-line has an angle with the block-line
upon which it fronts different from that of every other lot-line,
and all dependent on some block-angle which is not recorded.
If it is not desirable to bisect the block by the line ab, its di-
rection may be chosen as desired, the distances along it are
fixed by the fronts on one and the angular divergence from
that side which is chosen, and the lot fronts on the other side
of the block must be correspondingly increased or diminished.
When alleys are laid out in a block so that the interior lines
are accessible, it is very rare that after the block is improved
these lines can be measured under the same conditions as the
fronts. If alleys are not laid out, the difficulties are usually
much greater. Location of lot-lines by angle from the front is
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CITY SURl^EVlNC. 413
undoubtedly the most uniform and workmanlike method avail-
able to the surveyor. Hence, distances on the rear lines of the
corner lots should be omitted from the record, if their presence
would leave any doubt as to which method of location is in-
tended. It is not customary, nor is it desirable, that lot-lines or
distances shoulci be determined upon the ground before record-
ing a subdivision, but they should be platted by a man who
knows at least the first principles of trigonometry, and has an
accurately measured basis for his work.
285. The Datum-plane. — Levels referred to a permanent
datum are needed as soon as it is apparent that the town is to
be a living reality and not simply a town on paper. The da-
tum is a matter of some importance, and should have a simple
relation to some natural feature of the locality which will re-
tain a vital interest so long as the town exists. There is an
individuality in town-sites which usually determines for each
case very definitely what is best. High-water mark indicating
the danjjer of overflow; the lowest available outlet for a
drainage system in a flat country ; the average sea- or lake-
level, as afifecting commerce ; these are often chosen and may
serve as examples. The datum selected has its value accu-
rately determined and marked by a monument as enduring
as the granite hills, or, if that is impossible, as near this stand-
ard as can be secured ; a block of masonry, with a single and
durable cap-stone firmly bolted to its place, and bearing the
datum, or a known relation to it, definitely marked and secured
from abrasion is certainly possible for all.
286. The Location of Streets for which the most econom-
ical and practical system of grades may be secured is to be
considered when the plat is being prepared. Grades are usu-
ally established from profiles taken along the centre lines of the
street to be graded. This method is direct and protects the
public fund, for the grade, which can be executed at minimum
cost, the street being considered by itself, can be determined
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414 SURVEYING.
from such a profile. The method fails from the fact that it
treats the fund raised by taxation as the sum total of the pub-
lie interest. Parties representing abutting property appear
before the legislative body which has final action and seek to
amend the recommendation of the engineer, claiming that in-
terests which should receive consideration are injured by the
grades proposed. It seems plain that whatever is recommend-
ed by the city's officer should have the moral weight which
attaches to an impartial consideration of all the interests which
the city fathers are bound to recognize. But this involves a
change of method. The contour map of the district involved
seems to offer some help toward a solution. Methods by
which a rapid approximation of the amount of cut and fill in-
volved in any proposed grade may be arrived at are discussed
in Chapter XIII., on the Measurement of Volumes.
287. Sewer Systems. — A well-devised sewer system
touches very closely the public health. The information
which is necessary in order to act intelligently involves, if
storm-water is to be provided for, the area and slopes of the
whole drainage-basin in which lies the area to be sewered.
This will enable a close approximation to be made of the work
required of the mains at the point of discharge. Each sub-
district involves its own problem. The most economical
method of reaching every point where drainage is necessary
is learned by studying the details of topography. Borings
along the lines of proposed work to determine the character of
the soil and the depth of the bed-rock are necessary in order
that contractors may bid intelligently. This species of under-
ground topography sometimes modifies the location fixed by
surface indications.
288. Water-supply. — The need of a water-supply fur-
nishes new work to the surveyor. The distance and elevation
of the source of supply, the topography of the country through
which aqueducts or mains must be brought, eligible sites for
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CITY SURVEYING. 4^5
reservoirs, with their relation in distance and elevation to all
points to be supplied, are to be furnished to the hydraulic
engineer. The datum-plane for these maps and that of the
town should correspond.
289. The Contour Map, which is so generally useful from
the time the town is first planned until public improvements
cease to be considered, if surveyed carefully at first, has no
need to be retraced each time such a map is useful. It had
best be drawn in sections of sufficient scale for a working-plan,
and so arranged that when adjacent sections are placed side
by side the contour lines will be continuous. If the contours
of the natural surface are drawn in india-ink, and the contours
showing the changes made by different kinds of public work
be drawn in some color, the map may give a great amount of
information without becoming confused.
METHODS OF MEASUREMENT.
290. The Retracing of Lines * comes with the private use
of lots or blocks and with the execution of public improve-
ments. The demand for this class of work comes not once
only, but many times, and never ceases while there is life and
growth. The changes to which these forces give rise furnish
the main demand for knowing along what lines growth may
proceed unchallenged. The man who first fences a lot in the
middle of an unimproved block can ill afford to risk being com-
pelled to move his fence for what a survey would cost. But
the first attempt to go over any part of a subdivision and
locate a lot-line raises the question, how nearly alike can a
surveyor measure the same distance twice, or how nearly alike
can two surveyors measure the same distance. If the distance
noted on the recorded plat was not measured correctly, the
resurvey must differ from it, or by chance make a mistake of
the same amount. The difference which appears by compar-
* See also Art. 194 in chapter on Land Surveying.
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41 6 SURVEYING.
ing results is not the error which exists in either the original
or the resurvey ; it may be more than either error, it may be
less, being the algebraic difference of the two errors. If there
is no difference it means that the work is uniform, and may be
correct, but both may also be in error a like -amount. It has
happened in the days of twenty-foot rods and in a city of con-
siderable size that every rod used by surveyors was too long.
The change to steel tapes has not set matters wholly right.
If a man compares steel tapes bearing the brand of the same
manufacturer and offered for sale in the same shop, he soon
ceases to be surprised at a very appreciable difference in
length.
291. Erroneous Standards.— -How long is a ten-foot pole
or a hundred-foot tape is a pertinent and fundamental ques-
tion. It cannot be ignored when deeds call for a distance
from some other point, as fixing the beginning-point of the
parcel conveyed. When the deed describes lot number — , as
shown on the recorded plat, there is a theory in accordance
with which uniformity is all that is required — a distribution of
the distance between monuments in proportion to the figures
of the record. Property is often laid out with a view to this
theory of surveying. So long as block-boundaries are definitely
marked, a degree of precision is very readily secured by this
method which is rarely attained when surveyors attempt to
measure standard distances. If the surveyor faithfully meas-
ures the block through and every time distributes what he
finds in proportion to the record, though his block distances
may not agree with the record or with themselves, the lot-lines
will be much more likely to be the same than if he measures
his record distance and stops at the lot. This method assumes
that the lots abut one upon another, and reach from one monu-
ment to the other. But if this be true, the distances noted
must often refer to some empirical standard peculiar to this
block and not to the United States standard established by
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CITY SURVEYING. 417
law. But the courts recognize no standard, so far as the
author knows, but that which is established by law. So that
when a surveyor comes to mark lot one, finds the corner of the
block, and drives his stake by measuring from it the distance
which the record assigns to lot one, it is hard to prove that he
has not measured according to the subdivision, although he
has given no thought to the distance which remains for the
other lots. But trouble begins right here, for the theory which
is correct for lot one cannot be very wrong for lot two ; con-
tinue the process to lots six and eight, and give to another sur-
veyor who has been doing the same kind of work at the other
end of the block an order to survey lot seven. A conflict in
this case is certain unless the surveyor who laid out the sub-
division, and each of the others since, knew the length of his
tape and knew how to measure.
292. Trae Standards.— The U. S. Coast and Geodetic Sur-
vey Department at Washington standardizes steel tapes for a
nominal fee, giving their exact lengths at a given temperature,
or the temperature at which the tape is standard. By means
of such a standard tape, a standard test bar may be set and
graduated, and used as a permanent standard of length. If
this bar be of iron or steel, then no attention need be given to
the temperature at the time of graduating it, or when subse-
quent comparisons of steel tapes are made with it, since both
will be at the same temperature. In this case the bar becomes
a standard at that temperature at which the original tape was
found to be standard, by the Coast Survey comparison. For
this reason it would be well to require the C. & G. Survey
authorities to give the true length of the tape at a given tem-
perature (as 60° F.) and for a given pull.
Where and how to construct a standard rod, and how to
care for it so that it may be permanently reliable, each indi-
vidual had best determine for himself. It should be fastened
in its place in such a manner that it can expand and contract
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4l8 SURVEYING,
freely, i.e., without any strain from its supports. If it is made
of separate parts, these should be so joined together that there
can be no lost motion between the pieces. The whole requires
protection from the weather and to be so supported that it
cannot be bent by a blow. The writer has solved this problem
for himself in the following way : Bars of tool steel one inch
wide and one fourth of an inch thick are joined, as shown in
the sketch, to make the desired length 50 feet -}-; the whole is
i
^BD GS
Fig. 105.
fastened to the office floor by screws which hold the middle
firmly, but each side of the middle the holes drilled for the
screws are slotted sufficiently to allow for any possible change
of temperature. The joints are so close that a light blow is
necessary to bring the parts to place; the screws were set
home and then withdrawn a little, so that they should not
cause friction with the floor. After the fastening was com-
pleted the standard marks were cut upon the rod.
293. The Use of the Tape. — It is one thing to have a
tape of correct length ; it is another thing to be able to use it.
In an improved town with curb-lines clear, perhaps the most
obvious method is by a measurement along the grade with the
same tension as that at which the tape is tested. It is then
necessary to correct for temperature and to note all changes of
grade, reducing the observed distance on each grade by the
versed sine of the inclination or by the formula given in Chap.
XIV. By this method the tape is supported for its entire length,
and it is practicable to use a tape two or three hundred feet
long to advantage provided there are enough assistants to
keep it from being broken. A difficulty arises in the use of
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CITY SURVEYING. 419
this method from the fact that the town is not made for the
convenience of surveyors, and curb-lines are not usually clear
where measurements are needed, but are obstructed by piles
of building material, bales of merchandise, etc., and in some
towns the streets are so dirty that the graduation could not be
seen long if a tape were used in this way ; it would also be so
covered with drying mud that it could not be rolled in the
box when out of use, hence would be frequently broken.
Tapes that are wound on a reel, and have no graduations to
speak of, could be used in the mud, but the other objections
mentioned would still make the method of very limited appli-
cation. It is further to be noted that the laying-out of the
town, which is the basis of all later work, has all to be done
before the streets are graded or the curbs set. This work
must be done by some other method.
The usual method is to keep the ends of the tape horizon-
tal by using a plumb at that end of the tape where the surface
is lowest, and often at both ends if the ground is so irregular
or so covered with brush and weeds that the tape must be
kept off the ground. The tape assumes a curved form, and
the horizontal distance is something less than the length of the
tape. There is also a tension in the tape which, on account of
the elasticity of the metal, somewhat increases its length. As
the tension increases the sag diminishes, hence there is a
degree of tension, such that its effect is equal and opposite to
the effect of the sag. Call this the normal tension. If a line is
measured with a pull less than the normal tension for the tape
used, the tape will sag too much and there will be a minus
error due to this excessive sag ; if the pull used exceeds the
normal tension, there will be a plus error due to this excess.
It the pull has been uniform the total error ip either case is
proportional to the length of the line ; but if the pull has not
been uniform the error has varied irregularly with each length
of tape and can most readily be calculated by retracing the line
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420 SUkVEVWG.
and using the proper tension. In practice the tape is tested
with a known tension, and a tension so much above the ** nor-
mal " is adopted for field use that its plus error is equal to the
plus error of the test.
294. To determine the "Normal Tension " in a tape sup-
ported at given intervals. The tape forms a catenary curve,
since it carries no load but its own weight and is of uniform
section.
Let P = horizontal tension (pull) ;
w = weight of a unit's length of tape ;
e = base of Naperian logarithms ;
s = length of curve from origin ;
/ = distance between supports ;
W = wl = weight of tape ;
X and J = horizontal and vertical coordinates, origin at low
est point ;
X = il for cases considered.
Then by mechanics,*
JP —^ _ ^£
y = — (e P -\- e p — 2),
2W
J> tux _ wx
and s — — (e'p — e ~p).
P
We observe (i), that if - is constant r and s are constant for
the same length of tape ; (2), if P be measured, say ten pounds,
* The discussion here given is rigid, but both the development and the evalu-
ation of the equations are laborious. Il the curve be assumed to be a parabola,
which 11 may when ihe sag is small, the development is much simpler. See the
treatment of this subject, Art. 344, Chapter XIV.— J. B. J.
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CITY SURVEYING.
421
as a working condition,^ and s will vary with the weight of
PI P
every tape used, hence ^ = - is the ratio which must be
constant ; (3), if the surveyor can keep y constant, the same
conditions keep s constant, and if ^ varies s must vary; (4), if
P 1VX
x{=i i/) varies, and — varies in the same ratio, then -p- is con-
stant, hence the parts of the equations in parenthesis are con-
P
stant and y and s vary as / and —
TABLES SHOWING NORMAL TENSION AND EFFECT OF
VARIABLE TENSION.
/ = 100 feet. X = 50 feet.
Sac.
Pull.
Rbsultants ±
P
w'
y-
— error.
P
IV'
Elonga-
tion
4- error.
1 Error in /
Error in looo ft.
1 ^t-
0.055
0.040
1 0.028
0.020
0.013
i ^-^7
0.002
1
+
-
+
800
900
1000
IIOO
1200
1300
1400
1500
1600
1800
2000
2400
ft.
1.56
1.39
1.25
1. 14
1.04
0.96
0.89
0.83
0.78
0.70
0.62
0.52
ft.
0.065
0.051
0.040
0.033
0.028
0.023
0.020
0.017
0.014
O.OII
0.009
0.007
8
9
10
II
12
13
14
15
16
18
20
24
ft.
O.OIO
O.OII
0.012
0.014
0.015
0 016
0.017
0.019
0.020
0.022
0.025
0.030
ft.
0.002
0.006
O.OII
0.016
0.022
ft.
0.55
0.40
0.28
0.20
0.13
0.07
0.02
ft.
0.02
0.06
O.Il
0.16
0.22
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4^^
^URVEYWG.
/=Sa'. x = a5'. i
Sag.
Pull.
Rbsultants ±
w'
400
500
600
700
800
900
1000
1 100
1200
1300
1400
1500
1600
1700
1800
y-
ft.
0.78
0.63
0.5a
0.45
0.39
0.35
0.31
0.28
0.26
0.24
0.22
0.21
0.19
0.18
0.17
— error.
ft.
0.033
0.020
0.0T4
O.OIO
0.007
0.006 i
0.004
0.004
0.004
0.003
0.003
0.002 '
1
0.002
0.002
O.OOI
P
Elonga-
tion
+ error.
Error in /
Error in 1000 fL
-
+
-
+
8
10
12
1
14
16
18
20
22
24
26
28
30
32
34
36
ft.
0.003
0.003
0.004
0.004
0.005
0.006
0.006
0.007
0.008 \
0.008 j
0.009
0.009
O.OIO
O.OII
O.OII
0.030
0.017
O.OIO
0.006
0.002
0.60
0.34
0.21
O.II
0.04
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
O.OIO
0.03
0.06
0.08
O.IO
0.12
0.14
0.16
0.18
0.20
Assuming values of — , the formulas are readily solved for
any assumed distance between supports and the results tabu-
lated ; seven-place logarithms are best for this work.
The 100' tape is chosen because it furnishes a ready means of
calculating a table for any other length of tape by a decimal
reduction of the errors, per 1000', in proportion to the length
P
desired, and tabulated with values of — reduced in the samt^
w
proportion. There are those who use the roo' tape free-hand,
with 16 to 20 pounds pull, and say they do the work uniformly
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CITY SURVEYING,
A^l
PL
In the ordinary formula for elongation, A = -prry"^ we have
the section ky a multiple of w. The foregoing tables are calcu-
lated from the value w = 3.4^. The tension in the tape P
'>^
ErrcTS for VXf6 feet.
& MXrio^Zf
017
/
l^
\m
\n
I
V
\
\
/
I
\
It
\^
i'
r
k
/
\?
^
%S
k
\
^
^
Sl
ti
^
\
r&
ill
ft^
■*
'
>
^
jjt
^
*.^
\
fti
vu
\^
s<^
r>«
>^
"^
\
«*»,
,t^
s
\
i^f.
V
1
^^
■*ii
/"
fop
*v.
^
'i-T,
•■*»
.^^
4/Vk-^
^^^^
*^*.^
f^r
"* <d
"**^
'
■—-
.^2P,
»^'
^
"^
^'
fov
i%
inJji.
—J
^
..^
""
•— '
— —
igce
i^
no
-
a
»
a
X)
«
»
10
00
12
00
14
00
le
00
u
00
^
JUU
2a
00
■"i
Fig. X06.
♦ i? is the modulus of elasticity in pounds to the square inch, and k is ths
area of the cross-section in square inches, L being given in the same denomin»
tion as A.
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424' SURVEYING,
differs from the horizontal tension P^ so thatP' = /'secant %
(/ = incHnation to the horizontal), a second difference which
is so small that it may be neglected. Let E = 275000CX) (see
PI X aPI
Chapter XIV.), hence ^^ = ^j^^^^;^, nearly.
The same facts for 1000 feet distance are shown in
Fig. 106. In the tables the plus and minus errors are shown
separately fo.r a single length of tape only, and combined
for icxx)' feet ; in the figure they are separated for the
whole distance and the resultants of the table are the vertical
intercepts between the curves (minus errors) and the straight
line (plus errors). The sag for a single length of tape and cor-
P
responding — is shown by dotted curved lines; these are
plotted to a reduced vertical scale which is shown at the right
of the sketch.
295. The Working Tension. — In using these tables it is
best to measure the sag until the necessary pull for the tape is
learned. When the ends of the tape are at a known elevation
above a level surface, a rule at the middle of the tape will
show whether the pull is right. The fore chainman should
learn to pull steadily, not w;th a jerk, as he sticks the pin. A
more emphatic statement than the figure itself is of the
worthlessness of an unsteady hand at the forward end of the
tape it would be hard to make. A consciously constant
pull, the same every time, is necessary for good work. To ob-
serve the sag is the surveyor's means, in the field, of knowing
that the work is being done. He soon learns to judge with
considerable accuracy whether the proper pull is constantly
maintained. The proper pull is determined by the tension at
which the tape is tested ; call this /. Then, having weighed
the tape, ^ = ^- . Seek the plus error from elongation for this
value of - . then find the same plus error between the curve for
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CITY SURVEYING. 42^
that length of tape and the straight line; the corresponding —
is right for field use.
For example, a 50' tape weighs six ounces, and the pull,
when tested, was five pounds; /. - = — -1 — = 666, and the
elongation = o'.o83. The curve for a 50' tape marked —
error from sag is distant from the line marked + error from
P
pull the same amount when -- = 1233. Whence /*= 1233
X iV "^ 50 = 9i pounds, and the sag = o'.25. When a tape
is to be suspended freely in use, the tension at the test,/, should
not be such that the working tension P will be so great as to
be impracticable ; but it is also to be noted that slight varia-
tions of pull do not affect the result as much, when the tension
is considerably above the normal, as the same variations would
affect it if the tension were at or below the normal.
296. The Effect of Wind. — A very moderate wind has a
marked effect on the sag of the tape ; the wind-pressure on the
surface of tape exposed increases the sag and gives it a diago-
nal instead of a vertical direction. The exposed surface of the
tape constantly changes, and this results in vibrations which
make it difficult to tell where either end of the tape is. The
effect of its action, which is a minus error, varies approximately
as the square of the length of tape exposed. The effect of
winding up part of the tape so as to use a shorter length is to
increase the use of the plumb, which is also affected by the
wind, and the result is a loss of a part or all that is gained. A
high working tension reduces the effect of the wind. But the
only way to eliminate this source of error is to cease from any
piece of work when the wind is so high that it cannot be done
as it should be done. There are estimates, topography, etc.,
which do not require a high degree of precision and which
can be done when other work cannot.
26
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426 SURVEYING.
2xfj. The EflFect of Slope.— When the tape is used with
its ends at different elevations, if it hangs freely its lowest
point would not be in the middle, but nearer the lower end.
The corrections for sag and pull still apply, however, with
inappreciable error, for all practicable cases. The normal
tension, therefore, remains the same as for a level tape. A
correction must now be made, however, for the grade, the
value of which is / vers. /, where / is the distance measured
along the slope, and i is the angle with the horizontal. The
measured distance is always too great by this amount*
The available means by which the tape may be kept level
are: (i) The judgment of two field-hands. (2) On difficult
lines, the presence of the surveyor standing at one side where
his position has some advantages. A distant horizon often very
sharply defines the horizontal. (3) Where streets are im-
proved, although it may be impracticable to measure along the
slope, the known fall per 100 feet will give the needed infor-
mation. (4) Where none of these methods are sufficient, test
the judgment by plumbing at different heights and correcting
the pin if necessary. These methods will eliminate the worst
errors ; but where it is necessary to measure lengths of five
or ten feet, and then plumb from above the head, the uncor-
rected remnant will be considerable, probably that due an
inclination of two per cent on the whole length of such
lines, with very careful work to get so near. This difference
in the character of .lines is to be taken into the account in
balancing the survey. Note that the resultant error is always
minus.
298. The Temperature Correction. — ^The temperature of
the tape at the time when the work is done affects the result.
This is not the temperature in the shade that day, nor the
♦This question is fully discussed in Art. 347, Chapter XIV., where the cop
rcction is found in terms of the difference in elevation of the two ends. — J. B. J.
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CITY SURVEYING. 4:7
reading at the nearest signal station, but is the tempera-
ture out on the line, under the conditions which exist there.
A grass-covered slope, descending away from the sun, will
often show at the same time as much as twenty or thirty
degrees lower temperature than a bare hillside inclining
toward the sun. The thermometer is needed with the work.
If the co-efiicienl of expansion is not known, use 0.0000065
for 1° F.
It is very desirable in a city-surveyor's work that he be able
to apply his corrections at once while in the field. If he goes
out to measure any given distance, he must be able to fix his
starting-point and drive his stake at the finish. If the weather
is hot or cold, he knows what it differs from the temperature
at which his tape is tested, and applies the correction at once
to the whole distance. He watches that the pull is right,
that the tape is kept horizontal, that the work stops when
the wind is too severe, and that the checks show the desired
accuracy.
299. Checks. — Every piece of work should be carried on
till it checks upon other work, verifying its accuracy within
desired limits. This method ties up every survey at both ends.
In order to be prepared to do this expeditiously, the surveyor
should lay out general lines which should be joined into a sys-
tem embracing the town-site. The lines of leading streets and
the boundary-lines of additions give most valuable information
when made parts of such a system. This borders on the geo-
detic idea, but it will generally be impracticable to determine
the lengths of these lines by triangulation from a measured
base, for the stations can very rarely be so chosen that the
angles can be measured upon the whole length of the lines, or
the diagonals be observed at all. Still, the angles should be
measured upon the best base practicable. Permanent build-
ings and existing monuments showing the lines of intersecting
streets should be noted both for line and distance.
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428 SURVEYING.
MISCELLANEOUS PROBLEMS.
300. The Improvement of Streets involves— (i) The
estimation of the earthwork in the grading and shaping of the
street. (2) The location of the improvements along the lines of
the dedicated streets. City ordinances usually prescribe a cer-
tain width of sidewalks and roadway for each width of street.
(3) The location of improvements at the grade fixed by ordi-
nance. (4) The estimation of materials furnished by contractors
and used in the work. The position of monuments which will
be disturbed during the progress of the work is preserved by
witness-stakes driven beyond the limits of disturbance. When
this precaution is neglected it results in all sorts of angles and
offsets in the curb-lines, in cases where there is surplus or defi-
ciency in the original survey. Take a case improved one
block at a time, where the first block is established by record
distance from the right, the second block by record distance
from the left, and a third by running from this last point to
a point established at the end of the third block by measuring
again from the right, etc. The resulting lines of curb will not
give a suggestion of where the street was laid out. Some sur-
veyors are accustomed to replace from their witness-stakes the
monuments on the new grade. Such a practice is certainly
to be commended ; the small cost to the public treasury can
well be borne for the public good.
301, Permanent Bench-marks. — In order to secure accu-
racy and uniformity in elevations throughout a city, bench-
marks are established by running lines of levels radiating from
the directrix, and checking the work by cross-lines at conven-
ient intervals, these cutting the whole territory into small par-
cels, so that a standard bench-nriark will never be far from any
work which must be done.* This work is carried on as far as
* These various lines of levels will form a network, such as that shown in Art
407, Chapter XIV. , which should be adjusted once for all as described in that chapter
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CITY tiURVEYING. 429
grades are established, and generally as far as the city officers
are prepared to propose grades for adoption by ordinance.
There is a view of what constitutes or is essential to accurate
methods which would make every piece of work start from
first principles, so that it may not depend in any way upon er-
rors involved in work previously done. But work done on this
plan does not have to be extended very far before the results
will show plainly that there is a wide margin between the uni-
formity attained and the accuracy attempted.
302. The Value of an Existing Monument is based (i) on
the fact that it corresponds in character and position to a mon-
ument described on the recorded plat ; (2) on the custom to
place monuments upon the completion of a survey, and on the
supposition that this monument in question was set in pursu-
ance of such custom, although no monuments are noted on
the plat ; (3) on recognition by surveyors and owners of land
affected by it ; (4) on the knowledge that it was placed by a
competent surveyor at a time when data were accessible which
are not now in existence. The value of the evidence which
establishes or tends to establish the reliability of the monument
is primarily a question for the judgment of the surveyor. His
decision must be reviewed and defended before courts and ju-
ries when there is a difference of opinion.
The monument is valueless, or less valuable in all degrees,
when there is evidence that it has been disturbed. It some-
times happens that there is no better way to establish a corner
than to straighten up a stone which is leaning, but has not
been thrown entirely out of the ground. Inquiry often brings
out the fact that a stone, after being completely out of the
ground, has been reset either by agreement of owners adja-
and so one elevation obtained for each bench-mark. It is common for each
bench-mark in a city to have numerous elevations differing by several tenths oi
a foot, and all of about equal credence. — J. B. J.
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430 SURVEYING.
cent, or by the reckless individual who did the mischief, and is
still pointed out as the stone the surveyor set. As a recog-
nized corner such a stone has some value, i.e., it is to be sup-
posed that it is somewhere in the right neighborhood ; but if its
position can be verified from other points which have not been
disturbed the work should be retraced. If the original survey
was made in a careless way or the corner-stones were badly
set, they may help a careful man to come to an average line
which shall correspond with the recorded plat. Monuments
are sometimes moved or destroyed maliciously. It is wise for
a surveyor to test discreetly everywhere, but to be especially
careful where there has been quarrelling about lines.
There is a principle, recognized to some extent by the
courts, that the existing monument is the evidence of the orig-
inal survey, whether or not it is called for by the recorded
plat. The custom that the surveyor making the subdivision
and the plat for record shall set corner-stones is so far fol-
lowed that this is generally true, cases of accident, carelessness,
and mischief, and such cases as that mentioned below, being
somewhat exceptional, but many times very real. It is some-
times attempted to go a step further and affirm that the re-
corded plat is the record of the survey. This reverses the or-
der of events in most cases, the survey being made in order to
mark upon the ground the chief points of a plan already fixed
upon ; and as to all the main lines, the plat is not altered, how-
ever carelessly the survey may be made. There are subdivisions
where no monuments were set and where no certain evidence
is in existence of how or where the original survey was made,
or whether any survey was made at all, and yet there is a re-
corded plat. A surveyor being called upon to make a survey
of some parcel in such a subdivision, sets stones in order to se-
cure recognition for his theory of the proper location. If he
does his work carefully he undoubtedly does the public a ser-
vice. Can any amount of ignorance of when or why these
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CITY SURVEYING, 43 ^
stones were set ever make them evidence of the original sur-
vey? In other cases some monuments may be in existence,
but more would be convenient, — points are determined from
existing monuments in accordance with the recorded plat and
stones are set. Another surveyor may feel a little nervous
about manufacturing this sort of evidence of the original sur-
vey, or more likely, may think it too much trouble and a dam-
age to the business, for the more doubt the more work for the
surveyor, so drives his stake. Then comes the owner who,
desiring to secure a permanent corner, digs a hole about the
stake without taking offsets, throws it out, and sets in a stone
— an existing monument I This is no fancy .sketch, nor are
such facts so very rare. The young man who thinks he would
like to be a surveyor, but has no eyes nor ears for facts like
these, had better turn his attention to some other business.
Surveying is an art — not an exact science.*
303. The Significance of Possession. — Possession has a
value in re^stabKshing old lines where all monuments have
disappeared. It is a species of perpetuating testimony of their
positions. The average of a series of improvements will often
give a very close determination of where the corner must have
stood. The practised eye accustomed to sharply defined lines,
every lot having very nearly its right quantity, which are cus-
tomary where lines are well established, will notice at once the
irregular possession, — gaps between houses, vacant spaces
between fences and houses, too little for use, too much for
ornament, which may be seen where lines are in doubt and
every man expects the next surveyor to make a conflicting
survey. Like the men of the present, most men in the past
have preferred to be right — have made efforts to be right —
have employed surveyors ; we can judge where these men in
* Consult Judge Cooley's paper on the Judicial Functions or the Surveyor,
Appendix A, and also Art. 194 in chapter on Land Surveying, and Appendix H.
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432 SURVEYING,
the past worked from by seeing where their works are. The
legal principle has a bearing here, that " he who would sue to
dispossess another must first show a better- title.** The sur-
veyor who attempts to dispute possession must show better
evidence than possession of the right location of the lines he
is employed to retrace.
304. Disturbed Corners and Inconsistent Plats. — The
work of testing a corner that probably has been disturbed has
many points of likeness to the work of reestablishing corners
that have disappeared altogether. The recorded plat is in all
cases the basis of the work. When it records the results of a
survey it is to be presumed that the surveyor endeavored to
do accurate work ; hence his work, if not absolutely correct,
was probably uniform. Lines which are shown by the plat
as straight lines are to be retraced as straight lines. Lines in-
volve less liability to error than measurements, and are first to
be considered. Determine as many points as possible by
straight lines between existing monuments. • Then test the
measurements along the extreme lines and the streets which
are the basis of the subdivision. If the measurements between
undoubted corners agree with the plat so closely, or if they
differ so uniformly that the presumption of accurate work
is justified, corners that are out of line or out of proportionate
distance have the burden of proof against them. He who
would claim for them authority must show that they have not
been disturbed, and that they are consistent with some ra-
tional location. If there was no original survey, that fact is no
excuse for careless work at a later time ; there is always some
place to begin. The case when the recorded plat does not
agree with itself presents more difficulties, such as the follow-
ing: (1) The lines do not give the same points as the distance;
(2) The distances disagree among themselves; (3) The monu-
ments disagree with both lines and distances impartially, or
agree with one and disagree with the other, while the general
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CITY SURVEYING, 433
character of the work negatives the supposition that they were
ever carefully set. The object to be sought is not to perpet-
uate forever the blunders of the original survey, but to seek
the most rational adjustment of all the evidence, so that all parts
may be located with a minimum of conflict, and so that no
one shall be able to prove your survey wrong, i.e., show a
more reasonable location for any part. A consultation of
surveyors before too many conflicting interests have developed
is often advantageous.
305. Treatment of Surplus and Deficiency.*— It is gen-
erally a simpler problem to determine in which block differences
of measurement, whether surplus or deficiency, belong than it
is to know what to do with them in the matter of lot-location.
There has never been any theory invented for the treatment
of either surplus or deficiency which is able to stand the test
of the courts against all combinations of circumstances. A
few suggestions with the more probable limitations are all the
help that can be offered : every case must be investigated for
itself, (i) A distribution of the whole front in proportion to
the record distances meets general approval, at least in cases of
surplus, until it comes in conflict with possession. This is just
the time when an owner of ground wants to know what his
rights are, and it is also the time when no surveyor can tell
him. A compromise, or the verdict of a petit jury, which
passes foreknowledge, are the chief alternatives. The courts
say that he who would sue for possession must show a better
title. An examination shows that each has a better title
than any other to so much ground as the plat assigns to his
lot, but that no one has a better title than any other one to
any part of the surplus. The surveyor does wisely to take
note of possession and make, if he can, such a location as is in
accordance with the record, and yet not in conflict with posses-
sion. When this is not possible, let the map and certificate of
survey be made in such a way that they are simply a state-
* See also Art. 196 in chapter on Land Surveying.
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434 SURVEYING.
ment of the facts. It is not a surveyor's business to decide
legal questions or give judgment in ejectment. (2) Because
a suit for surplus will not lie, it has been thought that he who
first took possession of the surplus would be secure if he were
only careful to take it so that every other one might have his
ground. Trouble with this view arises because it is not possi-
ble to locate the surplus. When one man has appropriated all
there is in the block, and the rest but one have appropriated
each his proportionate share, then com^s the last man. The
more surplus in the block the more he is deficient ; he wants his
ground, and he finds it easier to sue'the one man than the twenty.
Perhaps, in order to be sure of a case, he had better sue them all.
The cases which arise in practice take on an infinite variety of
complications and are not usually so simple as these described.
(3) The fact is, that the idea that a subdivision ought to
have a little surplus is irrational. The work should be so
close to the standard that the surveyor who retraces the lines
would testify : ** According to the best of my knowledge and
belief, there is neither surplus nor deficiency there. In retrac-
ing my own work, which is carefully executed, I observe as
great discrepancies as any which I find in this subdivision, and
I conclude that the small difference which I observed in this
case was as likely to have been an error in my own work as to
attach to the subdivision." (4) Deficiency would seem to be
easier to deal with than surplus ; for when the last man has not
his ground he has a valid claim against the original owner for a
rebate on the purchase-price. But the burden of the difficulty
in this case falls on the surveyor. When a man brings his
deed and asks a survey of lot 9, while 8 and 10 are unsold and
lots I to 7 are already in possession, he leaves lot 8 its ground
and the deficiency in lot 10. Suppose it turns out that lot 10 is
next sold, and that the surveyor reports it deficient, the seller,
when waited on, may reply, ** I have not sold more ground in the
block than I owned ; the surveyor has made a mistake in locat
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CITY SURVEYING. 435
ing lot 9.** This liability attaches to every location which is
made before every lot, between the one located and one corner
of the block, is sold. (5) It is practicable for the original
owner to so write his deeds as to locate surplus or deficiency.
By beginning all deeds at the record distance from one street
and continuing this uniformly through the block, the differ-
ence goes in the lot farthest from the starting-point ; or he
may continue the process up to any line which he may choose,
and work from the other end of the block in deeding the re-
maining lots ; then the difference falls upon the line chosen
and falls to the share of the lot abutting upon that line which
is last deeded. But to approve this method is to affirm the
practicability of absolute accuracy in work. No one can tell
how small a difference may cause trouble.
306. The Investigation and Interpretation of Deeds * for
the use of the land-surveyor, dealing with the harmony or
conflict of the descriptions, is entirely a different work from
that of the investigator of titles, which deals with the legal
completeness of the conveyance. In the older parts of a town
the deed of the present proprietor frequently does not give
information sufficient to fix the correct location. The key
may lie in some boundary in an early deed referring to a still
earlier conveyance of adjacent property. Or the earlier deeds
may give clearly defined locations, while the latter ones
say " more or less*' at every point. In some cases the deeds
are in such a condition that it is impossible to tell what they
mean until it is known what the possession is. Skill in this
work can only come after considerable experience ; local prac-
tices must largely determine what is necessary.
307. Office Records. — The surveyor's office when well
planned is so arranged that no item of information which
promises to be useful shall be lost. The customary methods
of indexing, and of block-plats for keeping notes, do not take
a very firm hold on general lines or the connections between
♦ See also Art. 194, Chapter VII,
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43^ SURVEYING.
subdivisions; they fail, in fact, in that part of the work which
has the most vital relation to efforts at future improvement.
It is advisable to add to the block-plats and indexes a general
atlas of the whole town for office use, at a scale of say loo'
to the inch, so that an area nearly half a mile square may
appear on the open pages. Such an atlas may show the notes
of the general lines and their angles, the base-line measure-
ments, the relation of subdivisions to one another, and a
variety of other information which it is difficult to pick out in
the widely scattered field-notes which first gathered the in-
formation, and which, with their larger scale, the block-plats
are not well adapted to show in a connected form.
There are filed in connection with deeds many plats which
do not appear on the record plat-books of the recorder's
office ; these need to be indexed, or, better, abstracted for
office use.
The field-notes, when prepared for the surveyor's use in
the field, should show in an accessible and portable form all
the information which the office contains and which is rele-
vant to the survey in hand. Labor spent beforehand in a
thorough preparation of accessible information is labor saved.
308. The Preservation of Lines after the monuments
have disappeared is accomplished by means of notes on build-
ings, marks and notes on curbing, paving, fences, etc. Notes
on buildings describe not only the character of the building,
but the particular part noted, so that another man, years
afterward, using the same note would have no doubt of the
identity of the part. In a growing town the work of keeping
up the notes goes on without ceasing, — buildings are remod-
elled or rebuilt, streets reconstructed, destroying old marks.
The old becomes the new so constantly that the surveyor
who would preserve the information which he already has
must be constantly employed at the work of renewal. There
is no place either in the street or out of it where the surveyor
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CITY SURVEYING, 437
can place his mark and say to all comers, " Touch not." It
follows that whenever it is necessary to use any mark, about
the permanence of which there can be a shadow of a doubt,
the permanence of the mark must be shown by some prac-
ticable test ; it is careless to assume it.
309. The Want of Agreement between Surveyors arises
from differences of information or of judgment, and in a less
degree from differences of skill. These are all just as human
elements as the lawyer deals with in his work. Testimony is
affected by the interests of those who speak, and the judg-
ment varies with the temperament of the individual. Per-
haps one of the most difficult lines for a surveyor to draw is
that which separates his confidence in his own skill in retrac-
ing a survey which was confessedly inaccurate, from his re-
liance on testimony which is evidently biassed as to the posi-
tion or disturbance of monuments, and other facts which
may help him to form a correct judgment. Errors in execu-
tion may be kept within such limits that work which shows
differences in closing of I in 5000 should be retraced, and the
average observed differences in one surveying party's work
will not exceed i in 20000. Two sets of men working to
reach the same standard may err in opposite directions, so
that differences between two surveyors may reasonably be
expected to be somewhat larger than either would tolerate in
his own work.
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CHAPTER XIII.
THE MEASUREMENT OF VOLUMES.
310. Proposition. — The volume of any doubly-truncated
prism or cylinder^ bounded by plane ends, is equal to the area of a
right section into the length of t/ie element through t/ie centres of
gravity of the bases ^ or it is equal to the area of eitlier base into
the altitude of the element joining the centres of gravity of the
bases y measured perpendicular to that base.
Let ABCD, Fig. 107, be a cylinder, cut by the planes OC
and OB, the unsymmctrical right section EF being shown in
plan in EF. Whatever position the cutting planes may have,
if they are not parallel they will intersect in a line. This line
of intersection may be taken perpendicular to the paper, and
the body would than appear as shown in the figure, the line
of intersection of the cutting planes being projected at O.
Let A = area of the right section ;
^A = any very small portion of this area *
X = distance of any element from O ;
then ax = height of any element at a distance x from O.
An elementary volume would then be ax^A, and the total
volume of the solid ^^ ould be 2ax^A.
Again, the total volume is equal to the mean or average
height of all the elementary volumes multiplied by the area
of the right section.
The mean height of the elementary volumes is, therefore^
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THE MEASUREMENT OF VOLUMES,
439
'2axAA a2xAA
But
:2xAA
is the distance from O to the
A " A ' A
centre of gravity, G^ of the right section,* and a times this dis-
tance is the height of the element LK through this point.
Therefore, the mean height is the height through the centre of
o<c
<r:2l^_ ^ —
FlQ. X07.
gravity of the base, and this into the area of the right section
is the volume of the truncated prism or cylinder. The truth
of the alternative proposition can now readily be shown.
Corollary. When the cylinder or prism has a symmetrical
cross-section, the centre of gravity of the base is at the centre
of the figure, and the length of the line joining these centres
is the mean of any number of symmetrically chosen exterior
elements. For instance, if the right section of the prism be a
regular polygon, the height of the centre element is the mean
of the length of all the edges. This also holds true for paral-
lelograms, and hence for rectangles. Here the centres of gravity
* This is shown in mechanics, and the student may have to take it foi
grafted temporarily.
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440
SURVEYING.
of the bases He at the intersections of the diagonals ; and since
these bisect each other, the length of the line joining the in-
tersections is the mean of the lengths of the four edges. The
same is true of triangular cross-sections.
311. Grading: over Extended Surfaces. — Lay out the
area in equal rectangles of such a size that the surfaces of the
several rectangles may be considered planes. For common
rolling ground these rectangles should not be over fifty feet
on a side. Let Fig. 108 represent such an area. Drive p^s at
Fig. X08.
the comers, and find the elevation of the ground at each in-
tersection by means of a level, reading to the nearest tenth of
a foot, and referring the elevations to some datum-plane below
the surface after it is graded. When the grading is completed,
relocate the intersections from witness-points that were placed
outside the limits of grading, and again find the elevations at
these points. The several dififerences are the depths of excava-
tion (or fill) at the corresponding corners. The contents of
any partial volume is the mean of the four comer heights into
the area of its cross-section. But since the rectangular areas
were made equal, and since each corner height will be used as
many times as there are rectangles joining at that comer, we
have, in cubic yards.
4 4 4 4 4.
4 4 4
3 I
1 i i ,
"1 a 8 1
F =
4X27
[:s/., + 22A. + 32>i. + 4:sAj.
^i)
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THE MEASUREMENT OF VOLUMES.
441
The subscripts denote the number of adjoining rectangles
the area of each of which is A,
From this equation we may frame a
Rule. — Take each comer height as many times as there
are partial areas adjoining it, add them all together, and mul-
tiply by one fourth of the area of a single rectangle. T.iis
gives the volume in cubic feet. To obtain it in cubic yards,
divide by twenty-seven.
If the ground be laid out in rectangles, 30 feet by 36 feet,
then—— — = — X- = 10; and if the elevations be taken to
4 X 27 108
the nearest tenth of a foot, then the sum of the multiplied
comer heights, with the decimal point omitted, is at once the
the amount of earthwork in cubic yards. This is a common
way of doing this work. In borrow-pits, for which this method
is peculiarly fitted, the elementary areas would usually be
smaller.
In general, on rolling ground, a plane cannot be passed
through the four corner heights. We may, however, pass a
plane through any three points, and so with four given points
A
A,
A,
A.
^
\
A
/^ 6
A
A
\
N
A
A,
/'
\
3
3
3
A
A
A
\
8 3
Fig. 109.
on a surface either diagonal may be drawn, which with the
bounding lines makes two surfaces. If the ground is quite
irregular, or if the rectangles are taken pretty large, the sur-
veyor may note on the ground which diagonal would most
27
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442 SURVEYING.
nearly fit the surface. Let these be sketched in as shown in
Fig. 109. Each rectangular area then becomes two triangles,
and when computed as triangular prisms, each corner height
at the end of a diagonal is used twice, while the two other
corner heights are used but once. That is, twice as much
weight is given to the corner heights on the diagonals as to
the others. In Fig. 109, the same area as that in Fig. 108 is
j^ _h^ shown with the diagonals drawn which best fit
the surface of the ground. The numbers at
the comers indicate how many times each
height is to be used. It will be seen that
"•^i each height is used as many times as there are
Fig. iio. triauglcs meeting at that comer. To derive
the formula for this case, take a single rectangle, as in Fig,
1 10, with the diagonal joining comers 2 and 4. Let A be the
area of the rectangle. Then from the corollary, p. 423, we
have for the volume of the rectangular prism, in cubic yards,
F =
A (fh+K+Ji, ^ K + K + h;
2 X 27
O 0 /
= 6^('*' + ^^' + *' + 2^') (2)
For an assemblage of such rectangular prisms as shown in
Fig. 109, the diagonals being drawn, we have, in cubic yards,
y= 6"^ [2^ + 2^A, + 32A, + 42A, + s2A,
+ 62A. + 72A, + S2A;]; ... (3)
where A is the area of one rectangle, and the subscripts denote
the number of triangles meeting at a corner.
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THE MEASUREMENT OF VOLUMES. 443
As a check on the numbering of the corners, Fig. 109, add
them all together and divide by six. The result should be
the number of rectangles in the figure. In this case, if the
rectangles be taken 36 feet by 45 feet, or, better, 40 feet by 40.5
feet, then the sum of the multiplied heights with the decimal
point omitted is the number of cubic yards of earthwork, the
corner heights having been taken out to tenths of a foot.
The method by diagonals is more accurate than that by
rectangles simply, the dimensions being the same; or, for
equal degrees of exactness larger rectangles may be used with
diagonals than without them, and hence the work materially
reduced. In any case some degree of approximation is neces-
sary.
312. Approximate Estimates by means of Contours.—
{A) Whenever an extended surface of irregular outline is to
be graded down, or filled up to a given plane (not a warped or
curved surface), a near approximation to the amount of cut or
fill may be made from the contour lines. In Fig. 1 1 1 the full
curved lines are contours, showing the original surface of the
ground. Every fifth one is numbered, and these were the con-
tours shown on the original plat. Intermediate contours one
foot apart have been interpolated for the purpose of making
this estimate. The figures around the outside of the bound-
ing lines give the elevations of those points after it is graded
down. The straight lines join points of equal elevation after
grading ; and since this surface is to be a plane these lines are
surface or contour lines after grading. Wherever these two
sets of contour lines inteisect, the difference of their elevations
is the depth of cut or fill at that point. If now we join the
points of equal cut or fill (in this case it is all in cut), we ob-
tain a new set of curves, shown in the figure by dotted lines,
which may be used for estimating the amount of earthwork.
The dotted boundaries are the horizontal projections of the
traces on the natural surface of planes parallel to the final
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444
SURVEYING.
graded surface which are uniformly spaced one foot apart ver-
tically. These projected areas are measured by the planimeter
and called Ax^ At, At, etc. Each area is bounded by the
dotted line and the bounding lines of the figure, since on these
bounding lines all the projections of all the traces unite, the
slope here being vertical. For any two adjoining layers we
have, by the prismoidal formula* as well as by Simpson's one-
third rule,
V,., = ^{A, + 4At + At\ (I)
where A is the common vertical distance between the pro-
jected areas.
* For the demonstration of the prismoidal formula see Art. 314.
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THE MEASUREMENT OF VOLUMES, 445
For the next two layers we would have, similarly,
^3-5 = f(^. + 4^* + ^.); (2)
or for any even number of layers we would have, in cubic
yards,
^= ^-^(^> f 4^. + 2^. + 4^4 + 2^. + . . . . a:), (3)
3 A 2/
where n is an odd number, h and A being in feet and square
feet respectively.
(JB) Whenever the final surface is not to be a plane, but
warped, undulating, or built to regular outlines like a fortifi-
cation, a reservoir embankment, or terraced grounds, a differ-
ent method should be employed.
In the former method the areas bounded by the dotted
lines were areas cut out by planes parallel to the final plane
surface, passed one foot apart vertically. But since the map
shows only the horizontal projections of these planes, these pro-
jections, multiplied by the vertical distance between them,
would give the true volumes.
When the final surface is not to be a plane, proceed as fol-
lows : First make a careful contour map of the ground. Then
lay down on this map a system of contour lines, corresponding
in elevation to the first set of contours, but in a different
colored ink, which will accurately represent the final surface
desired. This second set of contours would be a series of
straight lines if a regular surface, composed of plane faces, was
to be constructed, but would be curving lines if the ground
were to be brought to a final curving or undulating surface.
The closed figures bounded by the two sets of intersecting
contours of the same elevation are horizontal areas of cut
or fill, separated by the common vertical distance between
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446
SURVEYING,
contours. The volumes here defined are oblique solids
bounded by horizontal planes at top and bottom, and are a
species of prismoid. The volume of one of these prismoids is
found by applying the prismoidal formula to it, finding the end
areas by means of a planimeter, and taking the length as the
G60
Fig. Ilia.
vertical distance between contours. If the contours be drawn
close enough together, then each alternate contour-area may be
used as a middle area, and the length of the prismoid taken at
twice the vertical distance between contours ; or the volume
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THE MEASUREMENT OF VOLUMES, 447
«
may be computed by either of the formulas (12), (13), (14), or
(15) of Appendix C, where the A*s would here become the end
areas and / the vertical distance between contours.
Example : Let it be required to build a square reservoir on
a hillside, which shall be partly in excavation and partly in
embankment, the ground being such as shown by the full con-
tour lines in Fig. iii^.*
The contours, for the sake of simplicity and brevity, are
spaced five feet aparjt. The top of the wall, shown by the full
lines making the square, is 10 feet wide and at an elevation of
660 feet. The reservoir is 20 feet deep, with side slopes, both
inside and outside, of two to one, making the bottom elevation
640 feet, and 20 feet square, the top being ico feet square on
the inside. The dotted lines are contours of the finished
slopes, both inside and out, at elevations shown on the figure.
The areas in fill all fall within the broken line marked abode
f g h i ky and the cut areas all fall within the broken line
m^ivkQd a b c d e f g 0, These broken lines are grade lines.
The horizontal sectional areas in fill and cut are readily traced
by following the closed figures formed by contours of equal
elevation, thus —
At 640 foot level sectional area in fill \s p s t.
« 650 " " " " " Imnuvx L
a 6jQ a a u u CUt is I 2 3 » X.
The other areas are as easily traced. In the figure the lines
have all been drawn in black. In practice they should be
drawn in different colors to avoid confusion.
This second method should be used«in all cases where the
graded area is^ considerable and the final relief form is not a
plane. If the contours be carefully determined and be taken
* This figure is taken from a paper describing the method by Prof. William
G. Raymond.
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448 SURVEYING,
near enough together, the method will give as accurate results
as may be obtained in any other way. The volume may be
computed by eq. (3) of this article, where the areas are the
horizontal sectional areas bounded by contours of equal ele-
vation, and h is the vertical distance between contours.
When these methods are used for final estimates, the con-
tours should be carefully determined, and spaced not more
than two feet apart on steep slopes and one foot apart on low
^slopes.
313. The Prismoid is a solid having parallel end areas,
and may be composed of any combination of prisms, cylinders,
wedges, pyramids, or cones or frustums of the same, whose
bases and apices lie in the end areas. It may otherwise be
defined as a volume generated by a right-line generatrix mov-
ing on the bounding lines of two closed figures of any shapes
which lie in parallel planes as directrices, the generatrix not
necessarily moving parallel to a plane director. Such a solid
would usually be bounded by a warped surface, but it can
always be subdivided into one or more of the simple solids
named above.
Inasmuch as cylinders and cones are but special forms of
prisms and pyramids, and warped surface solids may be divided
into elementary forms of them, and since frustums may also
be subdivided mto the elementary forms, it is sufficient to say
that all priomoids may be decomposed into prisms, wedges,
and pyramids. If a formula can be found which is equally
applicable to all of these forms, then it will apply to any com-
bination of them. Such a formula is called
314. The Prismoidal Formula.
Let A = area ofthe base of a prism, wedge, or pyramid ;
A^ A^j A^ = the end and middle areas of a prismoid, or of any
of its elementary solids ;
h = altitude of the prismoid or elementary solid.
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THE MEASUREMENT OF VOLUMES. *449
Then we have,
For Prisms,
l*or Wedges,
V^hA^\{A,^AA^^A^ (I)
hA h
V=''f = ^{A, + 4A„ + A,) (2)
For Pyramids,
V=^f = l{A,+4A„ + A,). .... (3)
Whence for any combination of these, having all the common
altitude A, we have
F = g(A+4^«+^,), (4)
which is the prismoidal formula.
It will be noted that this is a rigid formula for all prismoids.
The only approximation involved in its use is in the assump-
tion that the given solid may be generated by a right line
moving over the boundaries of the end areas.
This formula is used for computing earthwork in cuts and
fills for railroads, streets, highways, canals, ditches, trenches,
levees, etc. In all such Cases, the shape of the figure above
the natural surface in the case of a fill, or below the natural
surface in the case of a cut, is previously fixed upon, and to
complete the closed figure of the several cross-section areas
only the outline of the natural surface of the ground at the
section remains to be found. These sections should be located
so near together that the intervening solid may fairly be as-
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450* SURVEYING.
sumed to be a prismoid. They are usually spaced lOO feet
apart, and then intermediate sections taken if the irregularities
seem to require it.
The area of the middle section is never the mean of the
two end areas if the prismoid contains any pyramids or cones
among its elementary forms. When the three sections are
similar in form, the dimensions of the middle area are always
the means of the corresponding end dimensions. This fact
often enables the dimensions, and hence the area of the middle
section, to be computed from the end areas. Where this can-
not be done, the middle section must be measured on the
ground, or else each alternate section, where they are equally-
spaced, is taken as a middle section, and the length of the
prismoid taken as twice the distance between cross-sections.
For a continuous line of earthwork, we would then have, iu
cubic yards,
V= ^^{A,+4A,+2A,+4A,+2A,+4A, . . +^,), . (i)
where / is the distance between sections in feet. This is the
same as equation (3), p. 445. Here the assumption is made
that the volume lying between alternate sections conforms
sufficiently near to the prismoidal forms.
315. Areas of Cross-sections.— In most cases, in practice
at least, three sides of a cross-section are fixed by the conditions
of the problem. These are the side slopes in both cuts and
fills, the bottom in cuts and the top in embankments, or fills.
It then remains simply to find where the side slopes will cut
the natural surface, and also the form of the surface line on the'
given section. Inasmuch as stakes are usually set at the points
where the side slopes cut the surface, whether in cut or fill,
such stakes are called slope-stakes, and they are set at the time
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THE MEASUREMENT OF VOLUMES. 45 1
the cross-section is taken. The side slopes are defined as so
much horizontal to one vertical. Thus a slope of i^ to i means
that the horizontal component of a given portion of a slope-
line is i^ times its vertical component, the horizontal com-
ponent always being named first. The slope-ratio is the ratio
of the horizontal to the vertical component, and Is therefore
always the same as the first number in the slope-definition.
Thus for a slope of i^ to i the slope-ratio is l^.
316. The Centre and Side Heights. — The centre heights
are found as follows : Place in one column of the note-book
the surface elevation of the ground at the centre stake, as
given in the level book. Then take off from the profile the
elevation of the points of change of grade only, and compute
the elevation of grade at each station, from the known dis-
tance and grade. Place these elevations of grade in a column
alongside the first. Then take the differences and put in a third
column as the centre heights. These centre heights, together
with the width of base and side slopes in cuts and in fills, are the
necessary data for fixing the position of the slope-stakes. When
these are set for any section as many points on the surface
line joining them may be taken as desired. In ordinary rolling
ground usually no intermediate points are taken, the centre
point being already determined. In this case three points in
the surface line are known, both as to their distance out from
the centre line and as to their height above the grade line.
Such sections are called ** three-level sections," the surface lines
being assumed straight from the^ slope-stakes to the centre
stake.
317. The Area of a Three-level Section.
Let d and d' be the distances out, and
h and h! the heights above grade of right and left slope*
stakes, respectively ;
D the sum of ^/and d\ c the centre height, r the
slope-ratio, w the width of bed.
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452
SURVEY JKG.
Then the area ABCDE is equal to the sum of the four tri-
angles AEw, BCw, wCD, and wED. Or,
^ =
(^+^')^+('^ + A')f
.... (I)
This area is also equal to the sum of the triangles FCD and
FED^ minus the triangle AFB. Or,
. ( , w\ D vf , .
^ = U + -— ) (2)
FlO. 119.
Equation (2) can also be obtained directly from equation
(i) by substituting for h and h in (i) their values in terms of
2
d and w^ k=i , and then putting D^ d^ d'. Equation
(2) has but two variables, c and D, and is the most convenient
one to use.
318. Cross-sectioning. — It will be seen from Fig. 112 that
there are three elevations to be determined above (or below)
grade, and two distances out to be determined. A regular
line of levels is carried, checking on all pre-established benches.
At each position of instrument from which slope-stakes are to
be set, the ** height of instrument " is taken out, and the dif.
ference between this and the ** elevation of grade " figured for
the several sections, the " elevation of grade '* having been
taken from the profile, and already entered up for all stations
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THE MEASUREMENT OF VOLUMES. 453
and "grade points." By holding the rod at the section on
line and taking a reading, and subtracting this from the "height
of instrument," we obtain '* elevation of profile " at that sec-
tion ; subtracting this from the *' elevation of grade " for that
section in fills, and vice versa in cuts, we obtain the amount
of fill or cut, which can be roughly checked from the profile
itself, if desired. The Railroad Field Books usually give forms
for keeping these notes. If the ground were level transversely,
the distance to the slope-stakes would be
But this is not usually the case, and hence the distance out
must be found by trial. If the ground slopes \ ^^" >
from the centre line in a ] , \ the distance out will evidently
be more than that given by the above equation, and vice versa.
The rodman estimates this distance, and holds his rod at a cer-
tain measured distance out, d^. The observer reads the rod,
and deducts the reading from the height of instrument above
grade (or adds it to the depth of instrument below grade), and
this gives the height of that point, h^, above or below grade. Its
w
distance out, iSx^n^ should bed = A,r + -. If this be more than
the actual distance out,^/„ the rod is set farther out; if less, it
is moved in. The whole operation is a very simple one in prac-
tice, and the rodman soon becomes very expert in estimating
nearly the proper position the first time.
In heavy work — that is, for large cuts or fills, and for irregu-
lar ground — it may be necessary to take the elevation and dis-
tance out of other points on the section in order to better
determine its area. These are taken by simply reading on the
rod at the critical points in the outline, and measuring the dis-
tances out from the centre. The points can then be plotted
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454 SURVEYING.
on cross section paper and joined by straight or by free-hand
curved lines. In the latter case the area should be deter-
mined by planimeter.
319. Three-level Sections, the Upper Surface con-
sisting of two Warped Surfaces. — If the three longitudinal
lines joining the centre and side heights on two adjacent three-
level sections be used as directrices, and two generatrices, one
on each side the centre, be moved parallel to the end areas as
plane directers, two warped surfaces are generated, every cross-
section of which parallel to the end areas is a three-level sec-
tion. These same surfaces could be generated by two longi-
tudinal generatrices, moving over the surface end-area lines as
directrices. The surface would therefore be a prismoid, and
its exact volume would be given by the prismoidal formula.
The middle area in this case is readily found, since the center
and side heights are the means of the corresponding end di-
mensions.
The prismoidal formula, giving volumes in cubic yards,
^=6l^7(^- + 4^- + ^«)' .... (I)
could therefore be written ^
+ 4('. + |.)i>.]-j|^.. W
This equation is derived directly from eq. (i) above, and eq.
w
^2), p. 452. The quantity — is the distance from the grade-plane
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THE MEASUREMENT OF VOLUMES, 455
to the intersection of the side slopes, and is a constant for any
gfiven piece of road. It would have dififerent values, however,
in cuts and fills on the same line.
For brevity, let
w - hxf Iwc^ -^
Tr = '^' *"*^ T>07^=M:=^-
Here K is the volume of the prism of earth, loo feet long, in-
cluded between the roadbed and side slopes. It is first in-
cluded in the computation and then deducted. It is also a
constant for a given piece of road.
Equation (2) now becomes
where r„ and D^ are the means of c^c^ and D^D^, respectively.
This equation involves but two kinds of variables, c and Z>,
and is well adapted to arithmetical, tabular, or graphical com-
putation. Thus if / = 100 ; w = 18 ; and r = i^ ; then ^0=6;
and K = 200 ; and equation (3) becomes
V=m \kc. + 6) A + (^. + 6) A + 4(^„ + (>)D,:^ - 200 . (4)
If the total centre heights (to intersection of side slopes) be
represented by Cj, C,, and Ci, then eq. (3) becomes, in general,
where K = ^|^, and is independent of width of bed and of
slopes.
For any given piece of road, the constants AT, K\ and c^ are
known, and for each prismoid the C's and Z>*s are observed,
hence for any prismoid all the quantities in eq. (5) are known.
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456
SURVEYING,
320. Construction of Tables for Prismoidal Computa-
tion.— If a table were prepared giving the products K'CD for
various values of C and D^ it could be used for evaluating
equation (3), which is the same as equation (5). The argu-
ments would be the total widths {D^, and the centre heights
(Ci). Such a table would have to be entered three times for
each prismoid, first with C^ and D^ ; second with C, and Z>, ;
and finally with Cm and Z?^. If four times the last tabular
value be added to the sum of the other two, and K subtracted,
the result is the true volume of the prismoid.
VALUES OF €0 (= -) AND IC (= — ^-\ FOR VARIOUS WIDTHS
4 X 27^/
AND SLOPES.
Width
of
Road-
bed.
Slopes.
H tol.
Htol
% to 1.
1 10
1.
l^t
ol.
IH to 1.
l9itol.
2 tol.
Q
K
Co
10
K
»85
6.7
K
"3
Co
5 0
K
93
Co
4.0
K
74
c.
K
62
Co
2.9
K
53
Co
2.5
K
46
10
20
370
3.3
11
33
448
II
234
7-3
149
5 5
112
4 4
90
3.7
75
3»
64
2.8
56
1«
24
533
13
266
8.0
178
6.0
J33
48
107
4.0
89
3-4
76
30
67
13
36
636
»3
3»3
8.7
209
6.5
157
5.2
125
4.3
104
3-7
89
3-2
78
14
38
725
M
363
9-3
242
7.0
181
5.6
»45
4 7
121
4.0
X04
3-5
9»
15
30
833
»5
4»7.
10. 0
278
75
208
6.0
167
50
139
4 3
119
3.8
104
16
3a
948
t6
474
10.7
3.6
8.0
237
6.4
190
5-3
'58
4.6
135
4 0
118
17
34
1070
'7
535
'»-3
357
8.5
268
6.8
214
5-7
178
4.9
153
4.2
'34
18
36
1300
18
600
13. 0
400
9.0
300
7-2
240
6.0
200
5.1
171
4 5
no
19
38
'337
»9
668
12.7
446
95
334
7.6
267
6.3
223
4-4
191
4.8
167
SO
40
1481
20
740
13.3
494
10. 0
370
8.0
296
6.7
247
5-7
212
5.0
185
SI
«
1633
21
816
14.0
544
»° 5
408
8.4
327
7.0
272
6.0
233
5-2
204
2«
44 |i793
33
896
14.7 1 598
II. 0
448
8.8
359
7-3
299
6.3
256
5-5
224
S3
46 |i959
»3
980
»5-3 1 653
"•5
490
9.3
392
7 7
326
6.6 j28o
58
245
24
48 |3i34
24
1067
16.0 j 711
12.0
534
9.6
427
8.0
356
6.9 30s
6.0
267 j
S5
50 2315
25
1158
16.7 773
1
13.5
579
10 0
463
8.3
386
7« 33»
6.2
264;
S6
53 2504
26
1252
17.3 835
•30
626
10.4
501
8.7
4'7
7-4 358
6.5
3'3
27
54 2700
27 '350
18.0 ! 900
'3-5
675
TO. 8
540
9.0
450
7.7 '386
6.8
338
28
56 2904
28
1452
.8.7 968
14.0
736
II. 3
581
93
4B4
8.0 |4i5
7.0
363
29
58 '3ti5
29 »558
19-3 1038
M-5
779
.1.6
633
9-7
5'9
8 3 445
7.2
389
30
60 3333
30 J1667
ao 0 1 1 1 1
!
.50
833
13. 0
667 10.0
1
556
8.6 476
7.5 1417
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THE MEASUREMENT OF VOLUMES.
457
Table XL* is such a table, computed for total centre heights
from I to 50 feet, and for total widths from i to 100 feet.
In railroad work neither of these quantities can be as small as
one foot, but the table is designed for use in all cases where
the parallel end areas may be subdivided into an equal number
of triangles or quadrilaterals.
Note. — To use this table for mean-end-area volumes, take out volumes for
the end areas only, multiply their sum by three, and then subtract the volume K,
Example i. Three-level Ground having two Warped Surfaces. — Find the
volume of two prismoids of which the following are the field-notes, the width
of bed being 20 feet, and the slopes li to i.
28.9! o
Station 11.
Station 12.
Suiion 12 + 56.
+ 12.6
24.3
+ 18.6
+ 14.8
43-0
+ 22.0
40-3
+ 20.2
34-9
+ 9.5 +10.3 +16.6
From the table, p. 456, giving values of C% and K^ we find for w = 20^
and r = i^, Co = 6.7, and A'= 247.
The computation may be tabulated as follows:
Sta.
Width,
Height,
Partial Volume.
Volume of
Prismoid.
II
71.9
253
562
M
69.6
23.4
503 X 4 = 2012
12
67.4
21.5
447
3021 — 247
2774
M
63.3
19.2
374 X 4 = 1496
12 + 56
59.2
17.0
3"
.56(2254- 247)
1 124
* Modeled somewhat after Crandall's Tables, but adapted to give volumes
by the Prismoidal Formula at once instead of by the method of mean end areas
first and correcting by the aid of another table to give prismoidal volumes, as
Prof. Crandall has done.
f The numerators are the distances out, and the denominators are the heights
above grade, + denoting cut and — fill.
28
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.158
SURVEYING,
Entering the table (No. XI.) for a width of 71 and a height of 25, we find
548, to which add 7 for the 3 tenths of height, and 7 more for the 9 tenths in
width, both mentally, thus giving 562 cu. yds. for this partial volume. Simi-
larly for the width 67.4, and height 21.5, obuining 447 cu. yds. The correspond*
ing result for the middle area is 503, which is to be multiplied by 4, ihus givmg
2012 cu. yds. The sum of these is 3021 cu. yds., from which is to be subtracted
the constant volume JC, which in this case is 247 cu. yds., leaving 2774 cu. yds.
as the volume of the prismoid.
The next prismoid is but 56 feet long, but it is taken out just the same as
though it were full, and then 56 hundredths of ibe resulting volume taken.
The data for the 12th station is used in getting tnis result without writing it
again on the page.
Example 2. Five-level Ground having four Warped Surfaces. — Find the
volume of a prismoid of which the loUowmg are the field*notes, the width o|
bed being 20 feet, and the slopes li to i :
II.
28.9
+12.6
27.1
la. -r
15.0
+ 12.0
0
+18.6
20.0
+21.0
43.0
+22.0
12.5
0
18.5
40.3
^11.4 +12.0 +14.8 +19.6 +20.2
This is the same problem as the preceding, with intermediate heights
added.
To compute this from the table, it is separated into three prismoids, as shown
in Fig. 113,
--\ .
Let ABDGCFE be the crofiS-scction. This may be separated into the triangle
A he, and the two quadrilaterals BCGD and ACFE, The area of the triangle is
\cw. That of the right quadrilateral is, from Art. 184, p. 209.
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THE MkASUREMENT OF VOLUMES,
459
Similarly the area of the left quadrilateral is i {c'-h)[dk ) +iVA .
The total area of the section then is
/# = jUr - Ji){^k -"-)■¥ k'iTk^cw^- kdk + i^c-h)\d^ - ^ j |.
(I)
If the interior side elevations be taken over the edges of the base, then
w w
dk and dh both become zero, and the first and last terms disappear.
Or if the centre and extreme side heights are the same, these terms go out.
Experience shows that these terms can usually be neglected without material
error. If they are retained, each partial volume will be composed of five terms,
while if they are neglected there will be but three. The signs of these terms also
must be carefully attended to. When the interior side readings are taken over the
edges of the base, therefore, this equation t>ecomes
^=i(>tVA +<T«f + M)
(2)
The tables are well adapted to compute the prismoidal volume for five-level
sections by either of these formulae. Thus, if the adjacent section also has five
points determined in its surface, its area may be represented by an equation similar
to one of these, and from these end-area data mean values may be found for the
corresponding middle-area points, and the volumes taken out as before. In this
case the prism included between the road-bed and side- slopes, whose volume is A',
is not included, and hence its volume is not to be deducted from the result. The
computation by table XI. of equation (i) would be as follows :
Sta.
..j...
k'.
A-
c.
d,.
k.
dk-
A.
Partial Volumes.
Total
Volume.
M
12.628.9
13. 0 38.0
i
12.0
12. 0
13 .0
15.0
'3.8
13.5
18.6
16.7
14.8
20.0
19.2
X8.5
21.0
20.3
19. 6
43.0
4X.6
40.3
32. 0
2X.I
20.2
+9 + 108 + 114-1-279—10= 500
4(+6+io4 + io2 f 26o-i2)=i840
+3 + 100+ 90 + 242—13= 433
3762
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460
SURVEYING.
The use of the table is the same as before. First take out from the table the
volume corresponding to (c — h')ld' k ), which when evaluated for section 11
is (18.6 — 12.6) (15.0— 10) = 6.0 X 5.0. This is positive, and the volume corre-
sponding to a depth of 6.0 feet and a width of 5.0 feet is 9 cubic yards. Proceed
^o evaluate the remaining terms of eq. (i) in a similar manner, the last term
coming out negative. The dimensions of the mid section are the means of the
corresponding end dimensions, as before. If one end-area is a three level section
and the next a five-level section, the included prismoid is computed as a five-level
prismoid, the vanbhing points in the three-level section corresponding to the
interior side elevations on the five-level section being indicated in the field. Par-
tial stations, or prismoids, are first computed as though they were loo feet long
(for which the table is constructed), and then multiplied by their length and divided
by 100 as before.
If equation (2) may be used, the work is shortened very much. The columns
in h\ tfhtdk, and h, may be omitted, and there will also be but three terms in
each partial product. Thus, if sections 11 and 12 had been taken with the interior
elevations, each 10 feet from the centre line, we might have had something as
follows :
28.9
10. o
10. o
430
+ 12.6 +15.4 -1-18.6 -1-19.8 +22.0
12.
g7«i
10. o
40- 3
-II. 4 -1-12.5 +14.8 +17.4 +20.2
The computation then, by eq. (2), would have been :
Su.
-^A.
k'.
e.
k.
^k-
Partial Volumes.
Total
Volume.
11
M
12
28.9
28.0
27.1
15.4
14.0
12.5
18.6
16.7
14.8
19.8
18.6
17.4
43.0
41.6
40.3
137 -» 114 + 263 = 514
4 (121 -1- 102 + 239) = 1848
104 -1- 90 + 2IS = 409
2771
By this method the computation of a five-level section is little more trouble
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THE MEASUREMENT OF VOLUMES,
461
than that of a three-level seaion, and yet the intermediate points taken at a dis*
w
tance of — from the centre, are apt to increase the accuracy considerably on
ordinary rolling ground.
321. Three-level Sections, the Surface divided into
four Planes by Diagonals. — If the surface included between
two three-level sections be assumed to be made up of four
planes formed by joining the centre height at one end with a
side height at the other end sec-
tion on each side the centre line
(Fig. 114), these lines being called
diagonals, an exact computation of
the volume is readily made without
computing the mid-area. Two diag-
onals are possible on each side the
centre line but the one is drawn
which is observed to most nearly
fit the surface. They are noted in
the field when the cross-sections are
taken.
The total volume of such a prismoid in cubic * yards is
V = -^^— \{d^ + ^i>i + (^ + ^>, + Z>C -f UC
o X 27 L
where c^. h^, and hi are the centre and side heights at one sec-
tion and dx and dl the distances out, c^ A,', A,, d^, and d^l be-
Fig. 114.
♦ This volume is made up of six pyramids, three on each side of the centre
plane, these having their apices in the side heights to which the diagonals are
drawn, and their bases in the opposite cross-section, in the central vertical plane,
and in the roadbed respectively. The student should derive the formula.
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462
SURVEYING.
ing the corresponding values for the other end section. C and
C are the centre heights, H and /T the side heights, and D
and ly the distances out on the right and left diagonals.
Although this formula seems long, the computations by it are
very simple. Thus let the volume be found from the following
field-notes for a base of 20 feet and side slopes i^ to i*
A,
22
+ 8\
0
+ 8\
47-5
+ 25
34
+ 16
+ 4
\,6
+ 4"
The upper figures indicate the distances out and those
below the lines the heights, the plus sign being used for cuts.
The computation in tabular form is as follows :
Stau
d.
A.
c.
A'.
df.
dArd'.
id-\-dric.
DC.
lya.
Z
23
8
8
25
47.5
69.5
556
....
. . • •
a
34
16
hx
JI
4
+ 7/'
4
= 24
= 12
i6
50.0
aoo
88
128
88
128
^:eks = 65 X 10
= 650
6)162200
2
t7 ) 27033
lOOI
cu. yard
Is.
The great advantage of the method consists in the data
all being at hand in the field-notes.
Hudson's Tables* give volumes for this kind of prismoid.
* Tables for Computing the Cubic Contents of Excavations and Embank-
ments. By John R. Hudson, C.E. John Wiley & Sons, New Jfork, 1S84.
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IHE MEASUREMENT OF VOLUMES. 463
They furnish a very ready method of computing volumes wheu
this system is used.
322. Comparison of Methods by Diagfonals and by
Warped Surfaces. — Although the surveyor has a choice of
two sets of diagonals when this method is used, the real surface
would usually correspond much nearer the mean of the two pairs
of plane surfaces than to either one of them. That is, the
natural surface is curved and not angular, and therefore it is
probable that two warped surfaces joining two three-level sec-
tions would generally fit the ground better than four planes,
notwithstanding the choice that is allowed in the fitting of the
planes. More especially must this be granted when the truth
of the following proposition is established.
Proposition : The, volume included between two three4evel
sections having their corresponding surface lines joined by
warped surfaces, is exactly a mean between the two volumes
formed between the same end sections by the two sets of planes re-
sulting from the two sets of diagonals which may be drawn.
If the two sets of diagonals be drawn on each side the
centre line and a cross-section be taken parallel to the end
areas, the traces of tk e four surface planes on each side the
centre line on the cutting plane will form a parallelogram,
the diagonal of which is the trace of the warped surface on
this cutting plane. Since this cutting plane is any plane par-
allel to the end areas, and since the warped surface line bisects
the figure formed by the two sets of planes formed by the
diagonals, it follows that the warped surface bisects the volume
formed by the two sets of planes. The proposition will there-
fore be established if it be shown that the trace of the warped
surface is the diagonal of the parallelogram formed by the
traces of the four planes formed by the two sets of diagonals.
Fig. 115 shows an extreme case where the centre height is
higher than the side height at one end and lower at the other.
Only the left half of the prismoid is shown in the figure. The
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464
SURVEYING.
cutting plane cuts the centre and side lines and the two diago-
nals in efgh on the plane, and in e^f'g'h' on the vertical
projection. For the diagonal c^d^ the surface lines cut out are
e'f and /'A'. For the diagonal c^d^ they are e'g' and g'h\
For the warped surface the line cut out is e'h\ this being an
tfw<?i)
Fig. 115.
element of that surface. It remains to show that e'f'h'g' is a
parallelogram.
Since the cutting plane is parallel to the end planes all the
lines cut are divided proportionally. That is, if the cutting
plane is one «*** of / from ^„ then it cuts off one «*^ of all the
lines cut, measured from that end plane. But if the lines
are divided proportionally, the projections of those lines are
divided proportionally, and hence the points e\f,h\g' divide
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THE MEASUREMENT OF VOLUMES. 465
the sides of the quadrilateral ^/, c^\c^\d^^ proportionaliy. But
it is a proposition in geometry that if the four sides of a quad-
rilateral, or two opposite sides and the diagonals, be divided
proportionally and the corresponding points of subdivision
joined, the resulting figure is a parallelogram. Therefore ef'h
g' is a parallelogram, and e'k is one of its diagonals and hence
bisects it. Whence the surface generated by this line moving
along r,r, and d^d^ parallel to the end areas bisects the volume
formed by the four planes resulting from the use of both di-
agonals on one side the centre line. Q. E. D.
It is probable, therefore, that the warped surface would
usually fit the ground better than either of the sets of planes
formed by the diagonals. Furthermore, the errors caused by
the use of the warped surface (Table XI.) are compensating
errors, thus preventing any marked accumulation of errors in
a series of prismoids.* There are extreme cases, however,
such as that given in the example, Fig. 1 14, which are best
computed by the method by diagonals.
323. Preliminary Estimate from the Profile.— If the
cross-sections be assumed level transversely then for given
width of bed and side slopes, a table of end areas may be pre-
pared in terms of the centre heights. From such a table the
* The two methods here discussed are the only ones that have any claims to
accuracy. The method by '* mean end areas," wherein the volume is assumed
to be the mean of the end areas into the length, always gives too great a volume
(except when a greater centre height is found in connection with a less total
width, which seldom occurs), the excess being one half of the volume of the
pyramids involved in the elementary forms of the prismoid. This is a large «rror
even in level sections, and very much greater on sloping ground, and yet
it is the basis of most of the tables used in computing earthwork, and in some
States it is legalized by statute. Thus in the example computed by Henck*s
method on p. 462 the volume by mean end areas is 1193 cu. yards; by the
prismoidal formula it is 1168 cu. yards, while by the method by diagonals it was
only looi cu. yards. This was an extreme case, however, and was selected to
show the adaptation of the method by diagonals to such a form.
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466 SURVEYING,
end areas may be rapidly taken out and plotted as ordinate^
front the grade line. The ends of these ordinates may then
be joined by a free-hand curve, and the area of this curve
found by the planimeter. The ordinates may be plotted to
such a scale that each unit of the area, as one square inch,
shall represent a convenient number of cubic yards, as looo.
The record of the planimeter then in square inches and* thou-
sandths gives at once the cubic yards on the entire length of
line worked over by simply omitting the decimal point. Evi-
dently the scale to which the ordinates are to be drawn to give
such a result is not only a function of the width of bed and
side slopes, but also of the longitudinal scale to which the pro-
file line is plotted. The area of a level section is
A ^wc-^-rc^f (i)
where w, Cy and r are the width of base, centre height, and
slope-ratio respectively.
Now if A = the horizontal scale of the profile, that is the
number of feet to the inch, and if one square inch of area is to
represent lOOO cu. yards, the length of the ordinate must be
hA Aiwc + rc')
y= = -^^ ' (2)
"^ 1000 X 27 27,000 "^ ^
If values be given to A, ze/, and r, which are constants for
any given case, then the value of jf becomes a function of c
only, and a table can be easily prepared for the case in hand.
Since ^ is a function of the second power of ^, the second dif-
ference will be a constant, and the table can be prepared by
means of first and second diflferences. Thus if c takes a small
increment, as i foot, then the first difference is
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THE MEASUREMENT OF VOLUMES.
467
But this first difference is also a function of c, and hence when
c takes an increment this first difference changes by an amount
equal to
h
t."y =
27000
2r,
(4)
which is constant. An initial first difference being given for a.
certain value of r, a column of first differences can be obtained*
by simply adding the ^"y continuously to the preceding sum.
With this column of first differences the corresponding column
of values oi y maybe found by adding the first differences con-
tinuously to the initial value oi y for that column.*
TABULAR VALUES OF ^ IN EQUATION (2) FOR w=2o, r=ii. AND
h = 400.
c
o.'o
O.'l
o.'3
o.'3
o.'4
0/5
o.'6
o.'7
o.'8
o.'9
in.
in.
in.
in.
in.
in.
in.
in.
in.
in.
0
0.00
0.03
0.06
0.09
0,I3
o.xs
0.19
0.33
0.25
0.38
T
•3a
•35
•39
•4a
•46
•49
■53
.57
.61
.64
a
.68
.73
.76
.80
.84
.88
.93
.96
x.oo
X.05
3
1.09
»»3
X.I7
1. 33
1.36
x.3«
«.35
X.40
x-45
X.49
4
1. 54
X.59
1.63
1.69
»-73
1.78
X.83
1.88
X.93
X.99
5
a.04
3.09
3.14
8.19
3.34
3.30
3.36
3.41
3.47
3-5a
6
3.58
3.63
3.69
a- 75
3. 80
3.87
3.93
a.98
304
3.X0
7
3.»6
3aa
3.38
3-35
3 41
3 47
3-54
3.60
3.66
3-73
8
3-79
3-86
3-92
3.99
405
4-'3
4.19
4.36
4-33
4.40
9
4 47
4-54
4.60
4.68
4-75
4.83
4.89
4^97
504
5. II
10
5.18
5.36
5.33
5-40
548
5.56
564
5.72
5-79
5.87
It
5.9s
6.03
6.10
6.18
6.26
6.35
643
6.51
6.59
6.67
Ta
6.76
6.84
6.9a
7.00
7.09
7.t8
7.36
7-35
7-43
7 5a
'3
7.61
7.70
7.78
7.86
796
8.05
8.14
8.33
8.33
8.41
14
8.50
8.60
8.68
8.77
8.87
8.97
9.06
9.16
9.35
9-35
»5
9-44
9.54
963
9-73
9.83
994
10.03
Z0.Z3
X0.33
10.33
16
10 -43
«o.53
10. 6a
10.73
10.83
10.94
1X.04
11.15
IX.35
"•35
17
XZ.46
11.56
11.66
11.77
1Z.88
13.00
13. 10
za.az
ia.31
13.43
t8
»2-53
ia.64
"•75
13 86
13.97
'309
1390
»3-3a
13 •4a
1354
»9
'3.65
'3-77
X3.87
1399
14.10
14 "3
M-34
'4-47
14-58
14.70
ao
14.81
M.93
15.04
15.16
15.39
x5-4a
15-53
15.66
XS.78
15.90
'' For a further exposition of this subject, see Appendix C.
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468 SURVEYING,
The preceding table was constructed in this manner, for
w = 20 feet, r = i4; and h = 400 feet to the inch.
324. Borrow-pits are excavations from which earth has
been " borrowed " to make an embankment. It is generally
preferable to measure the earth in cut rather than in fill, hence
when the earth is taken from borrow-pits and its volume is to
be computed in cut, the pits must be carefully staked out and
elevations taken both before and after excavating. The meth-
ods given in art. 311 are well suited to this purpose, or they
may be computed as prismoids by the aid of Table XL, if pre-
ferred. To use the table it is only necessary to enter it with
such heights and widths as give twice the elementary areas
(triangles or quadrilaterals) into which the end sections are
divided, and then multiply the final result by the length and
divide by 100. The table is entered for both end-area dimen-
sions and also the mid-area dimensions, four times this latter
result being taken the same as before.
325. Shrinkage of Earthwork.— Excavated earth first
increases in volume, when removed from a cut and dumped on
a fill, but it gradually settles, or shrinks, until it finally comes
to occupy a less volume than it formerly did in the cut. Both
the amounts, initial increase and final shrinkage, depend on
the nature of the soil, its condition when removed, and
the manner of depositing it in place. There can therefore
be no general rules given which will always apply. For
ordinary clay and sandy loanty dumped loosely, as from wheel-
barrows, the first increase is about one twelfth, and then
the settlement about one sixth of this increased volume, leav-
ing a final volume of about nine tenths of the original volume
in cut.
Embankments made with carts or wagons will shrink from
the first volume in fill from five to ten per cent, while wheel-
scraper work will shrink from one to five per cent, depending
on the condition of the material when moved and the weather
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rHE MEASUREMENT OF VOLUMES. 469
conditions during the progress of the work. One may
judge of the probable shrinkage by estimating the propor-
tion of voids which will probably be ultimately filled by
settlement.
For rock the permanent increase in volume is from 60 to
80 per cent, the greater increase corresponding to a smaller
average size of fragment.
326. Excavations under Water. — It is often necessary to
determine the volume of earth, sand, mud, or rock removed
from the beds of rivers, harbors, canals, etc. If this be done
by soundings alone, it is likely to work injustice to the con-
tractor, as he would receive no pay for depths excavated below
the required limit ; and besides, foreign material is apt to flow
in and partially replace what is removed, so that the material
actually excavated is not adequately shown by soundings
within the required limits. It is common, therefore, to pay
for the material actually removed, an inspector being usually
furnished by the employer to see that no useless work is done
beyond the proper bounds. The material is then measured in
the dumping scows or barges. The unit of measure is the
cubic yard, the same as in earthwork. There are two general
methods of gauging scows, or boats. One is to actually meas-
ure the inside dimensions of each load, which is often done in
the case of rock, and the other is to measure the displacement
of the boat, which is the more common method with dredged
material. When the barge is gauged by measuring its dis-
placement, the water in the hold must always be pumped down
to a given level, or else it must be gauged both before and after
loading and the depth of water in the hold observed at each
gauging. A displacement diagram (or table) is prepared for
each barge, from its actual external dimensions, in terms of its
mean draught. There should always be four gaugings taken
to determine the draught, at four symmetrically located points
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470 SURVEYING.
on the sides, these being one fourth the length of the barge
from the ends. Fixed gauge-scales, reading to feet and tenths
may be painted on the side of the barge, or if it is flat-bot-
tomed, a gauging-rod, with a hook on its lower end at the zero
of the scale, may be used and readings taken at these four
points. Any distortion of the barge under its load, or any
unsymmetrical loading, will then be allowed for, the mean of
the four gauge-readings being the true mean draught of the
boat.
To prepare a displacement diagram, the areas of the sur-
faces of displacement must be found for a series of depths uni-
formly spaced. This series may begin with the depth for no
load, the hold being dry. They should then be found for each
five tenths of a foot up to the maximum draught. If the boat
has plane vertical sides and sloped ends these areas are rec-
tangles, and are readily computed. If the boat is modelled to
curved lines, the water-lines can be obtained from the original
drawings of the boat, or else they must be obtained by actual
measurement. In either case they can be plotted on paper,
and their areas determined by a planimeter. These areas are
analogous, to the cross-sections in the case of railroad earth-
work, and the prismoidal formula may be applied for comput-
ing the displacement. Thus,
Let A^y A^y A^y /}„ etc., be the areas of the displaced water
surfaces, taken at uniform vertical distances A apart. Then
Cor an even number of intervals we have in cubic yards
If the total range in draught be divided into six equal por-
tions, each equal to A, then Weddel's Rule* would give a
' For the derivation of this rule see Appendix C.
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THE MEASUREMEbTT OF VOLUMES 47 ^
nearer approximation. With the same notation as the above
we would then have, in cubic yards,
F=f-^K + ^. + A + ^. + S(^.+^. + A) + ^.].. (2)
These rules are also applicable to the gauging of reservoirs,
mill-ponds, or of any irregular volume or cavity.
After the displaced volume of water is found, the corre-
sponding volume of earth or rock is found by applying a proper
constant coefficient. This coefficient is always less than unity,
and is the reciprocal of the specific gravity of the material.
This must be found by experiment. In the case of soft mud
it is nearly unity, while with sand and rock it is much more.
When rock is purchased by the cubic yard, solid rock is not
implied, but the given quality of cut or roughly-quarried rock,
piled as closely as possible. When rock is excavated, solid
I rock is meant. A measured volume of any material put into a
\\gauged scoww'iM give the proper coefficient for that material.
Thus if the measured volume V give a displacement of V,
V
then -pr = C* is the coefficient to apply to the displacement to
give the volume of that materiaL
Note. — The computation of " haul" does not properly come within the prov-
* ince of this work. A very excellent article on this subject, showing the use of
the '• mass curve" in earthwork, by Prof. Walter L. Webb, will be found in the
Proceedings of the Engineers Club of Philadelphia^ vol. xiv. (1897), p. 249, and
also in the Railroad Gazette for December 17, 1897, p. 885. Also a paper by Prof.
C. Frank Allen in Railroad Gazette of May 24, 1895, p. 325.
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Q. -n-^ «.. n.' .^v..^ ^^^.. /.v // f/ 'V- '
, l4 ] GEODETIC SURVEYING.* (J
327. The Objects of a Geodetic Survey are to accurately
determine the relative positions of widely separated points on
the earth*s surface and the directions and lengths of the lines
joining them ; or to accurately determine the ^z^^^/«/^ positions
(in latitude, in longitude from a fixed meridian, and in eleva-
tion above the sea-level) of widely separated points on the
earth's surface and the directions and lengths of the lines join*
ing them.
. In the first case the work serves simply to supply a skeleton
of exact distances and directions on which to base a more de-
tailed survey of the intervening country ; in the second, the re-
sults furnish the data for computing the shape and size of the
earth, in addition to their use in more detailed surveys.
It is usually desirable also to have some knowledge of the
latitude and longitude of the points determined in the first
case, but a very accurate knowledge of these would not be es-
sential to the immediate objects of the work.
In both cases the points determined form the vertices of a
series of triangles joining all the points in the system. One or
more lines in this system of triangles and all of the angles are
very carefully measured, and the lengths of all other lines in
the system computed. The azimuths of certain lines are also
determined, and, if desired, the latitudes and longitudes of some
of the points. From this data it is then possible to compute
the latitudes and longitudes of all the points in the system and
* See Appendix F for a discussion of many subjects considered in this chapter.
GEODETIC SURVEYING, 473
the lengths and azimuths of all the connecting lines. The
work as a whole is denominated triangulation.
The measured lines are called base-lines, the points deter-
mined are triangulation-stations, and those points (usually tri-
angulation-stations) at which latitude, longitude, or azimuth
is directly determined are called respectively latitude, longi-
tude, or azimuth stations. The latitude of a station and the
azimuth of a line are determined at once by stellar observations
at the point. The longitude is found by observing the differ-
ence of time elapsing between the transit of a star across the
meridian of the longitude-station and the meridian of some
fixed observatory whose longitude is well determined. An ob-
server at each station notes the time of transit across his merid-
ian, and each transit is recorded upon a chronograph-sheet at
each station. This requires a continuous electrical connection
between the two stations. This difference of time, changed
into longitude, gives the longitude of the field-station with ref-
erence to the observatory.
328. Triangulation Systems are of all degrees of magni-
tude and accuracy, from the single triangle introduced into a
course to pass an obstruction, up to the large primary systems
covering entire continents, the single lines in which are some-
times over one hundred miles in length.
The methods herein described will apply especially to what
might be called secondary and tertiary systems, the lines of
which are from one to twenty miles in length, and the accu^
racy of the work anywhere from I in SCXX) to I in 50,000. Al-
though the methods used are more or less common to all sys-
tems, yet for the primary systems, where great areas are to be
covered and the highest attainable accuracy secured, many
refinements, both in field methods and in the reductions, are
introduced which would be found useless or needlessly expen-
sive in smaller systems.
If it is desired to connect two distant points by a system
29
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474
SURVEYING,
of triangulation at the least expense, then use system I., shown
in Fig. ii6. This system is also adapted to the fixing of a
double row of stations with the least labor.
If such distant points are to be joined, or such double system
of stations established, with the greatest attainable accuracy,
then system III. should be used. This system is also best
adapted to secondary work, where it is desired to simplify the
work of reduction. Each quadrilateral is independently re-
duced.
If the greatest area is to be covered for a given degree of
accuracy or cost, then system II. is the one to use.
System I. consists of a single row of simple triangles, sys-
Fig. xi6.
tem II. of a double row of simple triangles or of simple tri-
angles arranged as hexagons, and system III. of a single row
of quadrilaterals. A quadrilateral in triangulation is an arrange-
ment of four stations with all the connecting hnes observed.
This gives six lines connecting as many pairs of stations, over
which pointings have been taken from both ends of the line.
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GEODETIC SURVEYING.
475
For the same maximum length of lines we have the follow-
ing comparison of the three systems : *
1
System.
Compositioo.
Distance
Covered.
No. of
Sta-
tions.
Total
Length
of
Sides.
Area
Covered.
No. of
Conditions.
I.
II.
III.
Equilateral triangles.
Hexagons.
Quadrilaterals (squares).
5
5-2
4-95
II
17
14
19
34
29
45
9
3-5
» — 2 = 9
5
2w - 4 = 28
Thus, for the same distance covered, the number of sta-
tions to be occupied and the total length of lines to be cleared
out are about one half more for systems II. and III. than for
system I. The area covered by system II. is twice that by
system I., but the number of conditions is much greater in
system III. than in either of the others. Since almost all the
error in triangulation comes from erroneous angle-measure-
ments, the results will be more accurate according to our
ability to reduce the observed values of the angles to their
true values. The " conditions" mentioned in the above table
are rigid geometrical conditions, which must be fulfilled (as
that the sum of the angles of a triangle shall equal i8o°), and
the more of these geometrical conditions we have, the more
neaily are we able to determine what the true values of the
angles are. The work will increase in accuracy, therefore, as
the number of these conditions increases, and this is why sys-
tem III. gives more accurate results than systems I. and II.
This will be made clear when the subject of the adjustment of
the observations is considered.
329. The Base-line and its Connections. — The line
whose length is actually measured is called the base-line. The
♦Taken from the U. S. C. and G. Survey Report for 1876.
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-47&
SURVEYING.
lengths and distance apart of such lines depend on the charac-
ter of the work and the nature of the ground. Primary base-
lines are from three to ten miles in length, and from 200 to
SkoiTovMT .
CHICAGO
BASE LINE SYSTEM
Scato 1:100.000
wm/tBam
600 miles apart. In general, in primary work, the distance
apart has been about one hundred times the length of the
base. Secondary bases are from two to three miles in length,
^BoKinm
Fig. xi8*
and from fifty to one hundred and fifty miles apart, the dis-
tance apart being about fifty times the length of base. Ter-
tiary bases are from one half to one and a half miles in length
* Taken from professional papers. Corps of Engineers U. S. Army, No. 24,
being the final report on the Triangulation of the United States Lake Survey.
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GEODETIC SURVEYING, 477
and from twenty-five to forty miles apart, the distance apart
being about twenty-five times the length of base.*
The location of the base should be such as to enable one
side of the main system to be computed with the greatest
accuracy and with the least number of auxiliary stations for a
given length of base. In flat open country the base may be
chosen to suit the location of the triangulation-stations in the
main system ; but in rough country some of the main stations
must often be chosen to suit the location of the base-line. In
Fig. 1 1 7 the location of the base-line is almost an ideal one,
being taken directly across one of the main lines of the sys-
tem. By referring to Fig. ii8 it will be seen that the line
Willow Springs — Shot Tower is one of the fundamental lines
of the main system, and the base is located directly across it.
Here the ground is a flat prairie, and the base was chosen to
suit the stations of the main system.
The station at the middle of the base is inserted in order
to furnish a check on the measurements of the two portions
as well as to increase the strength of the system by increasing
the number of equations of conditions. Sometimes it is neces-
sary to use one or more auxiliary stations outside the base
before the requisite expansion is obtained. Thus suppose the
stations Morgan Park and Lombard were the extremities of
the line of the main system whose length was to be computed
from this base, then the stations Willow Springs and Shot
Tower might have been occupied as auxiliary stations from
which the line Morgan Park — Lombard could be computed.
330. The ' Reconnaissance. — A system of triangulation
having been fixed upon, of a given grade and for a given pur-
* These intervals between bases are in accordance with the practice that
has hitherto been followed. The new method of measuring base-lines wiih a
steel tape, described in Art. 339, will probably change this practice by causing
more bases to be measured, leaving much shorter intervals to be cov^ed by
angular measurement.
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478 SURVEYING,
pose, the first thing to be done is to select the location of the
base-line and the position of the base-stationsT The base should
be located on nearly level groundQaind should be favorably sit-
uated with reference to the best location of the triangulation-
stations. These stations are then located, first for expanding
from the base to the main system, and then with regard to the
general direction in which the work is to be carried, and to the
form of the triangles themselves.
No triangle of the main system should have any angle less
than 30° nor more than 120°. Although small angles can be
measured just as accurately as large ones, a given error in a
small angle, as of one second, has a much greater effect on the
resulting distances than the same error in an angle near 90*^.
In fact, the errors in distance are as the tabular differences in a
table of natural sines, for given errors in the angles. These
tabular differences are very large for angles near 0° or 180°, but
reduce to zero for angles at 90°. The best-proportioned tri-
angle is evidently the equilateral triangle, and the best-propor-
tioned quadrilateral is the square. In making the reconnais-
sance the object should be to fulfil these conditions as nearly
as possible.
The most favorable ground for a line of triangles is a valley
of proper width, with bald knobs or peaks on either side. Sta-
tions can then be selected giving well-conditioned triangles,
with little or no clearing out of lines, and with low stations.
In a wooded country the lines must be cleared out or else very
tall stations must be used. In general, both expedients are re-
sorted to. Stations are built so as to avoid the greater portion
of the obstructions, and then the balance is cleared out.
So much depends on the proper selection of the stations in
a system of triangulation, as to time, cost, and final accuracy,
that the largest experience and the maturest judgment should
be made available for this part of the work. The form of the
triangles ; the amount of cutting necessary to clear out the
lines and the probable resulting damage to private interests;
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GEODETIC SURVEYING, 479
the height and cost of stations, and the accessibility of the
same ; the avoidance of all sources of atmospheric disturbance
on the connecting lines, as of factories, lime- or brick-kilns, and
the like, which might either obstruct the line by smoke or in-
troduce unusual refraction from heat ; the freedom from dis-
turbance of the stations themselves during the progress of the
work, and the subsequent preservation of the marking-stones —
these are some of the many subjects to be considered in de-
termining the location of stations.
It is the business of the reconnaissance party not only to
locate the stations, but to determine the heights of the same.
A station that has been located is temporarily marked by a
flag fastened upon a pole, and this made to project from the
top of a tall tree in the neighborhood. In selecting a new
station it is customary to first select from the map the general
locality where a station is needed, and then examine the region
for the highest ground available. When this is found, the
tallest trees are climbed and the horizon scanned by the aid of
a pair of field-glasses to see if the other stations are visible. If
no tree or building is available for this purpose ladders may
be spliced together and raised by ropes until the desired height
b obtained.
331. Instrumental Outfit. — ^The reconnaissance party re-
quires a convenient means of measuring angles and of determ-
ining directions and elevations. For measuring angles a pocket
sextant would serve very well, provided the stations are distinct
or provided distinct range-points in line with the stations may
be selected by the aid of field-glasses. A prismatic pocket-
compass will often be found very convenient in finding back
stations which have been located and whose bearings are known.
An aneroid barometer is desirable for determining approxi-
mate relative elevations. For methods of using it in such
work, see Chapter VI., p. 136. If to the above-named instru-
ments we add field-glasses, and creepers for climbing trees, the
instrumental outfit is fairly complete.
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480 SURV-EYJNG.
332. The Direction of Invisible Stations.— It often liap.
pens that one station cannot be seen from another on account
of forest growth, which may be cleared out. In such a case the
station may be located and the line cleared from one station or
from both, the direction of the Une having been determined.
This direction may always be computed if two other points
can be found from each of which both stations and the other
auxiliary point are visible. Thus in Fig. 119 let AB be the
line to be cleared out, and let C and D
be two points from which all the stations
may be sighted. Measure the two angles
at each station and call the distance CD
unity. Solve the triangle BCD for the
side BCy and the triangle ADC for the
side AC We now have in the triangle
ABC two sides and the included angle to
^'°- "9 find the other angles. When these are
found the course may be aligned from either A or B. It will
often happen that either C ox D ox both can be taken at regu-
lar stations. Of course a target must be left at either C ox D
to be used in laying out the line from A or B. The above is a
modification of the problem given in art. 1 10, p. 107. A use of
this expedient will often greatly facilitate the work.
333. The Heights of Stations depend on the relative
heights of the ground at the stations and of the intervening
region. If the surface is level, then the heights of stations
depend only on their distance apart. In any case the dis-
tance apart is so important a function of the necessary height
that it is well to know what the heights would have to be for
level, open country.
The following table* gives the height of one station when
the other is at the ground level, for open, level country:
* Taken from Report of U. S. Coast and Geodetic Survey for 18S2.
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GEODETIC SUJ^VEYJAC,
481
DIFFERENCE IN FEET BETWEEN THE APPARENT AND TRUE
LEVEL AT DISTANCES VARYING FROxM i TO 66 MILES.
Dis-
tance,
miles.
Difference in feet for— i
1
Dis-
tance,
miles.
Curvature.
Refraction.
Curvature :
and J
Refraction.
Curvature.
Refraction.
Curvature
and
Refraction.
I
0.7
O.I
0.6 1
34
771-3
108.0
663.3
2
2.7
0 4
2.3 1
35
817.4
114. 4
703.0
3
6.0
0.8
5.2
36
864.8
121. 1
743 7
4
10.7
I 5
9.2
37
913.5
127.9
785.6
5
16.7
2.3
14.4
38
963.5
134.9
828.6
6
24.0
3-4
20.6
39
1014.9
142. 1
872.8
7
32.7
4.6
28.1
40
1067.6
149-5
918. 1
8
42.7
6.0
36.7
41
II21.7
157.0
964.7
9
540
7.6
46.4
42
II77.O
164.8
1012.2
10
66.7
9-3
57.4
43
1233.7
172.7
1061.0
II
80.7
II. 3
69.4
44
1291.8
180.8
IIII.O
12
96.1
13.4
82.7
45
1351-2
189.2
1162.0
13
112. 8
158
97.0
46
1411.9
197.7
1214.2
14
130.8
18.3
112. 5
47
1474.0
206.3
1267.7
15
150. 1
21.0
129. 1
48
1537.3
215.2
1322. I
16
170.8
23.9
146.9
49
1602.0
224.3
1377.7
17
192.8
27.0
165.8
50
1668. I
233.5
1434.6
18
216.2
30.3
185.9
51
1735.5
243-0
1492.5
19
240.9
33.7
207.2
52
1804.2
252.6
1551.6
20
266.9
37-4
229.5
53
1874.3
262.4
1611.9
21
94.3
41.2
253-1
54
1945-7
272.4
1673-3
22
322.9
45.2
277-7
55
2018.4
282.6
1735.8
23
3530
49.4
303.6
56
2092.5
292.9
17996
24
384-3
53-8
330.5
57
2167.9
303-5
1864.4
25
417.0
58.4
358.6
58
2244.6
3142
1930.4
26
451. 1
63.1
388.0
59
2322.7
325.2
1997.5
27
486.4
68.1
418.3
60
2402 . I
336.3
2065.8
28
323.1
73.2
449-9
61
2482.8
347.6
2135.2
29
561.2
78.6
482.6
62
2564.9
359.1
2205.8 1
30
600.5
84.1
516.4
63
2648.3
370. 8
2277.5 1
31
641.2
89 8
551-4
64
2733.0
382.6
2350.4
32
683.3
95.7
587.6
65
2819. I
394.7
2424.4 i
33
726.6
101.7
624.9
66
2906.5
406.9
2499.6
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482 SURVEYING,
square of distance
Curvature =
mean diameter of earth *
Log curvature = log square of distance in feet — 7.6209807 ;
Refraction = ^nty where K represents the distance in feet,
R the mean radius of the earth (log R = 7.3199507), and m the
coefficient of refraction,* assumed at .070, its mean value, sea-
coast and interior.
-AT*
Curvature and refraction = (i — 2m) —n-
Or, calling A the height in feet, and K the distance in statute
miles, at which a line from the height A touches the horizon,
taking into account terrestrial refraction, assumed to be of the
same value as in the above table (070), we have
.7575' 1.7426-
The following examples will serve to illustrate the use of
the preceding table :
I. Elevation of Instrument required to overcome Curvature
and Refraction, — Let us suppose that a line, A to B, was 18
miles in length over a plain, and that the instrument could be
elevated at either station, by means of a portable tripod, to a
height of 20 or 30 or 50 feet. If we determine upon 36.7 feet
at A, the tangent would strike the curve at the distance rep-
resented by that height in the table, viz., 8 miles, leaving the
curvature (decreased by the ordinary refraction) of 10 miles to
be overcome. Opposite to 10 miles we find 57.4 feet, and a
*Sce discussion on refraction, Arts. 396-8.
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GEODETIC SURVEYING 483
signal at that height erected at B would, under favorable
refraction, be just visible from the top of the tripod at A^ or
be on the same apparent level. If we now add 8 feet to tripod
and 8 feet to signal-pole, the visual ray would certainly pass 6
(eet above the tangent point, and 20 feet of the pole would be
visible from A,
II. Elevations required at given Distances. — If it is desired
to ascertain whether two points in the reconnaissance, esti-
mated to be 44 miles apart, would be visible one from the
other, both elevations must be at least 278 feet above mean
tide, or one 230 feet and the other 331 feet, etc. This sup-
poses that the intervening country is low, and that the ground
at the tangent point is not above the mean surface of the
sphere. If the height of the ground at this point should be
200 feet above mean tide, then the natural elevations should
be 478 or 430 and 531 feet, etc., in height, and the line is
barely possible. To insure success, the theodolite must be
elevated at both stations to avoid high signals.
Since the height of station increases as the square of the
distance, it is evident that the minimum aggregate station
height is obtained by making them of equal height. Or, if
the natural ground is higher at one station than the other,
then the higher station should be put on the lower ground —
that is, when the intervening country is level. If, however,
the obstruction is due to an intervening elevation, the higher
station should be the one nearer the obstruction.
Sometimes a very high degree of refraction is utilized to
make a connection on long lines. Thus on the primary trian-
gulation of the Great Lakes three lines respectively 100, 93,
and 92 miles in length were observed across Lake Superior,
which could not have been done except that the refraction was
found sometimes to exceed twice its average amount. The line
from station Vulcan, on Keweenaw Point, to station Tip-Top
in Canada, was 100 miles in length. The ground at station
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4^4 Purveying.
Vulcan was 726 feet above the lake, and the observing station
was elevated 75 feet higher, making 801 feet above the surface
of the lake. The station at Tip-Top was 1523 feet above the
lake, the observing tripod being only 3 feet high. From the
above table we find that the line of sight from Vulcan would
become tangent to the surface of the lake at a distance of 37.4
miles, and that from Tip-Top at a distance of 51.5 miles, thus
leaving a gap of about eleven miles between the points of
tangency, for ordinary values of the refraction. If this inter-
val were equally divided between the two stations and these
raised to the requisite height, we would find from the table
that Tip-Top would have to be elevated some 340 feet and
Vulcan some 260 feet. Since this was not done, we must con-
clude that an occasional excessive value of the refraction was
sufficient to bend these rays of light by about these amounts
in addition to the ordinary curvature from this source. In
other words, the actual refraction when one of these stations
was visible from the other must have been more than double
Its mean amount.
The following is a synopsis of the heights of the stations
built for the observation of horizontal angles in the primary
triangulation of the Great Lakes :
Total number of stations* » 243
Combined height of stations 14,100 feet
Average height of stations 58 "
Average height of stations from Chicago to Buffalo 81.3 **
Number of stations less than 10 feet high .... 23
*• •* from 10 feet to 24 feet in height 18
•* 25 " 49 " " 50
" 50 •* 74 •* " 71
" 75 •• 09 " " 47
• 100 *• 109 " " 18
•• " '* no ** 119 '• *' 15
•* 120 •' 124 •* *• 2
♦Only stations built expressly lor the work are here included. Sometimef
buildings or towers were utilized in addition to these.
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GEODETIC SURVEYING.
485
The heights above given are the heights at which the in-
strument was located above the ground. The targets were
usually elevated from 5 to 30 feet higher.
The excessive heights of the stations from Chicago to
Buffalo are due to the country being very heavily timbered,
and the surface only gently rolling. In the vicinity of Lake
Superior they averaged only about 35 feet high, while from
Buffalo to the eastern end of Lake Ontario they averaged 51
feet in height. ^
334. Construction of Stations. — If it is found necessary
to build tall stations, two entirely separate structures must be
ju»a
M,BsMot
OBSCRVtNQ TRIPOD
Fig. 120.
erected, one for carrying the instrument and one for sustain-
ing the platform on which the observer stands. These should
have no rigid connection with each other. These structures are
shown in plan and elevation in Figs. 120 and 121. The inner
station is a tripod on which the instrument rests; this is sur-
rounded by a quadrangular structure, shown separately in ele-
vation to prevent confusion. Both structures are built entirely
of wood, the outer one being usually carried up higher than
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4^6
PURVEYING.
the tripod (not shown in the drawing), and the target fixed to
its apex. This upper framework serves also to support an
awning to shade the instrument from the sun. For lower sta-
tions a simpler construction will serve, but the observer's plat-
form must in all cases be separate from the instrument tripod.
The wire guys and wooden braces shown in Fig. 120 were not
used on the U. S. Lake Survey stations.
For stations less than about 15 feet in height the design
?r
\
/
/
/
"-^\
-^^ ^
.17^/t
OROUND PLAN Scale
Fig. 131.
aoo
shown in Figs. 122 and 123 may be used. Here the outer
platform on which the observer stands is entirely separate from
the tripod which supports the instrument. For ground stations
a post firmly planted serves very well, or a tree cut off to the
proper height. The common instrument tripod will seldom be
found satisfactory for good work. Sometimes extra heavy and
•stable tripods of the ordinary pattern have given excellent re-
suits.
335. Targets. — The requisites of a good target are that it
ihall be clearly visible against all backgrounds, readily bisected.
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GEODETIC SURVEYING.
487
rigid, capable of being accurately centred over the station, and
so constructed that the centre of the visible portion, whether
in sun or in shade, shall coincide with its vertical axis.
Fig. 12a.
It is not easy always to fulfil these conditions satisfactorily.
To make it visible against light or dark backgrounds, it is well
Fig. Z33.
to paint it in alternating black and white belts. For ready bi-
section it should be as narrow as possible for distinctness. This
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488
SURVEYING.
IS accomplished by making the width subtend an angle of from
two to four seconds of arc. Since the arc of one second is
three tenths of an inch for one-mile radius, an angle of four
seconds would give a target one tenth of a foot in diameter for
one-mile distances, or one foot in diameter for ten-mile dis-
tances. Something depends on the magnifying power of
the telescope used. The design shown in Fig. 124 will satis-
Fic. 124.
Fig. 125.
factorily satisfy the conditions as to rigidity and convenience
of centering. Of course it should stand vertically over the
station so that a reading could be taken on any part of its
height. The last condition is not so easily satisfied. If a
cylinder or cone be used the illuminated portion only will
appear when the sun is shining, and a bisection on this portion
may be several inches to one side of the true axis.
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GEODETIC SURVEYING, 489
The target is then said to present a phase, and corrections
for this are sometimes introduced. It is murh better, however,
to use a target which has no phase. If the target is to be read
mostly from one general direction, a surface, as a board, may
be used ; but if the target is to be viewed from various points
of the compass, then from those stations which lie nearly in the
plane of the target it would not be visible, from its width being
so greatly foreshortened.
In this case two planes could be set at right angles, one
above the other. One or both would then be visible from all
points, and since their axes are coincident, either one could be
used. The objection to this would be that the upper disk would
cast its shadow at times on the lower one, leaving one side in
sun and the other in shade, thus giving rise to the very evil it
is sought to eliminate. A very satisfactory solution of this
problem was made on the Mississippi River Survey by means
of the following device (Fig. 125): Four galvanized-iron wires,
about three-sixteenths inch in diameter, are bent into a circle of,
say, four inches in diameter, and soldered. To these four circles
are attached four vertical wires about one fourth inch in diam-
eter and four feet long, as shown in the accompanying figure.
All joints to be securely soldered, the size of the wire increas-
ing with the size of the target. The target is now divided into
a number of zones by stretching black and white canvas alter-
nately and in opposite ways between the opposing uprights,
making diametral sections. If there are more than two zones,
those marked by the same color should have the canvas cross-
ing in different ways, so that if one plane is nearly parallel to
any line of sight the other plane of this color will be nearly at
right angles to it. This target has no phase, is visible against
any background, and readily mounted. A wooden block may
be inserted at bottom, with a hole in the axis of the target.
This may then be set over a nail marking the station. The
target is held at top by wire guys leading off to stakes in the
.30
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490
SURVEYING.
ground. Such a target could be. mounted on top of the pole
shown in Fig. 124, if it should be found necessary to elevate it.
336. Heliotropes. — When the distance between stations
is such that, owing to the distance, the state of the atmosphere,
or the small size of the objective used, a target would appear
indistinct, or perhaps not be visible at all, the reflected rays of
the sun may be made to serve in place of a target. This limit-
ing distance is usually about twenty miles. Any device for
accomplishing this purpose may be called a heliotrope. In
Figs. 126 and 127 are two forms of such an instrument. That
Fig. ia6.
Fig. ia7.
shown in Fig. 126 is a telescope mounted with a vertical and
horizontal motion. This is turned upon the station occupied
by the observer, and is then left undisturbed. On the tele-
scope are mounted a mirror and two disks* with circular open-
ings. The mirror has two motions so that it can be put into
any position- Its centre is coincident with the axis of the
disks, in all positions. The mirror may be turned so as to
* The disk next lo the mirror is unnecessary.
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GEODETIC SURVEYING. 49 ^
throw a beam of light symmetrically through the forward disk,
in which position the reflected rays are parallel to the axis of
the telescope, and hence fall upon the distant point.
The heliotrope shown in Fig. 127 is to be used in conjunc-
tion with a single disk, which may be a plain board mounted
on a plank with the mirror. The silvering is removed from a
small circle at the centre of the mirror. The disk has a small
hole through it as high above its base as the clear space on the
mirror is above the plank. The operator points the apparatus
by sighting, through the clear spot on the mirror and the open-
ing in the disk, to the distant station. If the plank be fas-
tened in this position the attendant now has only to move the
mirror so as to keep the cone of reflected rays symmetrically
covering the opening in the disk, and the light will be thrown
to the distant station. ,'^i- '"^"^tv
Since the cone of incident rays subtends an>aff^le €f about 3 <> '
thirty-two minutes, the cone of reflected rays subtends the
same angle. The base of this cone has a breadth of about
fifty feet to the mile distance, or at a distance of twenty miles
the station sending the reflection is visible over an area in a
vertical plane 1 000 feet in diameter. The alignment of the
heliotrope need not, therefore, be very accurate. This align-
ment may vary as much as fifteen minutes of arc on either side
of the true line. This is nearly o.oi of a foot in a distance of
two feet. If the bearing, or direction, of the distant station is
once determined, it may be marked on the station by some
means within this limit, and a very rude contrivance used for
sending the reflected ray, or flash, as it is called. Thus, a mir-
ror and a disk with tbe requisite movements may be mounted
on the ends of a board or pole from five to twenty feet long,
and when this is properly aligned it serves as well as any other
more expensive apparatus. The hole in the disk should usually
subtend an angle at the observer's station of something less
than one seco»»d of arc, which is a width of three-tenths of an
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49^ sv/iVEVhyG,
inch to the mile distance. On the best work with large instru-
ments it should subtend an angle of less than one half a
second, the minimum effective opening depending almost
wholly on the condition of the atmosphere.*
Whatever form of heliotrope is used, an attendant is re-
quired to operate the apparatus. Evidently it can be used
only on clear days, whereas cloudy weather is much better
adapted to this kind of work, since the atmosphere then trans-
mits so much clearer and steadier an image.
The heliotrope can be used as a means of communication
between distant stations by some fixed code of flashing sig-
nals, and it has been so used very often with great advantage
to the work. The attendant on the heliotrope,^ usually called
a flasher, can thus know when the observer is reading his sig-
nals, when he is through at that station, and, in general, can re-
ceive his instructions from his chief direct from the distant
station.
337. Station Marks. — If the triangulation is to serve for
the fixing of points for future reference, then these points must
be marked in some more or less permanent manner. In this
case the station has been chosen with this in view, so that if
possible it has been provided that even the surface for a few
feet around the station shall remain undisturbed. To insure
against disturbance from frost or otherwise, the real mark is
usually set several feet underground. Many different means
are employed to mark these points. The underground mark
IS to serve only when the superficial marks have been dis-
turbed, there being always left a mark of some kind projecting
above ground. On the U. S. Lake Survey, "the geodetic
point is the centre of a i-inch hole drilled in the top of a stone
* Reflected sunlight has been seen a distance of sixty miles, through an
opening one inch in diameter, which then subtended an angle of but one eigh-
teenth of one second of arc at the instrument. This would require a very dear
atmosphere.
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GEODETIC SURVEYING, 493
two feet by six inches by six inches, sunk two and one-half
feet below the surface of the ground. When the occupation
of the station is finished, a second stone post, rising eight
inches above the ground, is placed over the first stone. Three
stone reference-posts, three feet long, rising about a foot above
the ground, are set within a few hundred feet of the station,
where they are the least likely to be disturbed. A sketch of
the topography within a radius of 400 metres about the sta-
tion is made, and the distances and azimuths of the reference-
marks are accurately determined."
When the station is located in natural rock a copper bolt
may be set to mark the geodetic point.
On the Mississippi River survey, stations had to be set on
ground subject to overflow. These were to serve both for
geodetic points and for bench-marks, both their geographical
position and their elevation being accurately determined.
Both the rank growth and the sedimentary deposits from the
annual overflows would soon obliterate any mark which was
but slightly raised above the surface. After much study given
to the subject, the following method of marking such points
was adopted: A flat stone eighteen inches square and four
inches thick, dressed on the upper side, has a hole drilled in
the centre, into which a copper bolt is leaded, the end project-
ing a quarter of an inch above the face of the stone. The
U S
stone is marked thus, ^ . j^, and is placed three feet under
ground. On this stone, and centred over the copper bolt, a^
cast-iron pipe four inches in diameter and five feet long isj
placed, and the dirt tamped in around it. The pipe is large
enough to admit a levelling rod. The top is closed with a cap,
which is fastened to the pipe by means of a bolt. The eleva-
tions of both the top of the pipe and of the stone are de-
termined.
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SURVEYING.
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GEODETIC SURVEYING. 49S
MEASUREMENT OF THE BASE-LINE.
338. Methods. — The methods formerly employed for the
measurement of primary and secondary base-lines have now
(1901) been generally abandoned in America. For these
former methods see Reports of the U. S. C. & G. Survey
for 1873, 1882, and Primary Triangulation of the U. S. Lake
Survey. The bases which are now employed by the engi-
neer corps of the U. S. Army are measured by means of 300-
or 500-ft. steel tapes. The U. S. C. & G. Survey have
adopted the Eimbeck ** Duplex " apparatus as shown in Figs.
128 and \2%a. This consists of two measuring-tubes, five
metres long, each containing two bars, one of steel and the
other of brass. As the expansion of brass is about one and
one-half times that of steel, the relative lengths of two such
bars, after a proper standardization, would furnish an index
of the common temperature of both of them. The greatest
source of error in all accurate length measurements has always
been the uncertainty in the temperature, and hence of the
length, of the measuring unit.* Mercurial thermometers can-
not be relied on to give the temperature of metallic bars
under rapidly changing temperatures, since these give only
the temperatures of their own bulbs. As the mass or the
cross-section of a thermometer bulb is small as compared to
that of a measuring-bar, the larger body responds more slowly
to atmospheric changes of temperature than the smaller bulb
of the thermometer, and hence in a changing temperature a
* M. Guillaume has discovered that an alloy of steel with thirty-five
or thirty six per cent of nickel has the unique property of not varying its
length appreciably for ordinary changes of temperature. This would make
an ideal material for measuring base-lines. See Bulletin de la Societe cC En-
couragement pour Industrie Nationale for March, i8g8. See also Johnson's
Materials of Construction (second and subsequent editions). To accom-
plish the same end, a steel bar packed in ice has been tried with ^ood
results. See Report U. S. C. & G. Survey for 1892.
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496
SURVEYING.
t
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GEODETIC SURVEYING. 496a
mercurial thermometer never records the mean temperature
of the measuring-bar, even though placed alongside and both
enclosed in metallic tubes with non-conducting coverings. On
the U. S. Lake Survey a combination of steel and zinc bars
was employed in the famous Repsold apparatus, but the
zinc was found to lag in its volumetric changes behind those
of the steel, and hence proved a very unsatisfactory metal for
the purpose, although its expansion was about two and one-
half times that of the steel. The latest and most successful
employment of a bimetallic base apparatus is the system
shown in Figs. 128 and 128^, which will now be described.*
There is shown in Fig. 128, page 494, a single tube,
mounted on two wooden tripods. In service this is aligned
by means of the telescope at the left end, and its vertical
angle is determined by means of the sector mounted on the side
at the centre of the tube. Two such tubes are used in the.
measurement, and the measuring-bars project at both ends.
These bars are brought into end-contact by means of thumb-
screws, as shown in Fig. 128^, page 496. These thumb-
screws move the bars bodily in the tube against the action
of spiral springs. The end contacts are made steel to
steel and brass to bass. The rear tube is then carried for-
ward and its rear end brought into contact with the forward
end of the stationary tube, and contact made again by moving
the bars in the forward tube. These contact-ends are agate
surfaces, one being a vertical plane and the other which meets
it a horizontal knife-edge. * The work proceeds in this way at
an average rate of some fifty or sixty tubes (250 or 300
metres) an hour, a maxinuim speed of eighty tubes having been
attained. The work may be done under an awning-frame,
which can be dragged forward on sled-runners by a team.
* For a full description of this apparatus and of the results which have
been obtained by it, see Rep. U. S. C. & G. Survey for 1897.
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49^^ SURVEYING.
Good results are obtained, however, by working in the open.
The relative position of the bars in the tube can be changed
by rotating i8o° the inner or ** reversing" tube, which carries
the bars, the outer or ** truss** tube remaining fixed.
Since the brass bar will in general be longer or shorter
than the steel bar (longer for temperatures above and shorter
for temperatures below the normal), it is clear that by always
bringing steel to steel and brass to brass in making the end-con-
tacts, one bar will continually gain upon the other. There is
an arrangement, however, for moving the brass bar forward
or back with reference to the steel whenever this deviation
reaches too large an amount, as five centimetres for instance.
There is a vernier attached to the steel bar at each end, which
reads upon a scale upon the brass bar, by means of which
this occasional adjustment can be determined and its amount
recorded in the notes. With these corrections applied there
will result two continuous measurements of the base, one by
the steel bar and one by the brass bar. Three mercurial
thermometers are placed in each tube, and all are read for
every contact ; these may or may not be given weight in com-
puting the length of the base. The relative lengths of the
entire base as measured by the steel-bar record and by the brass-
bar record give the key to the average temperature of both bars
for the entire base, provided the two bars be assumed to always
liave the same temperature. Since these bars are in reality
small tubes, of a relative thickness of metal to compensate for
relative specific heats and conductivities, and are both en-
closed in a double tubular covering, it has been found by trial
that they may be assumed to always have the same tem-
perature.
The apparatus is standardized by measuring with it a known
distance at two very different temperatures. This furnishes
data for computing the absolute (and hence the relative)
lengths of both bars, and their coefficients of expansion, and
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GEODETIC SURVEYING, ^97
hence their lengths at any temperature. Thereafter, given
the relative lengths of the bars, their temperatures and hence
their absolute lengths can be found. The ** known distance"
is usually a short base-line which has been measured by
means of a standard bar packed in ice to hold it at a constant
temperature.
The Steel Tape furnishes the most convenient, rapid, and
economical means for measuring any distance for any desired
degree of accuracy up to about one in three hundred thousand,
and if the most favorable times are chosen, an accuracy of i
in 1,000,000 may be attained. It is probable, therefore, that
all engineering measurements, even mcluding primary base-
lines, will yet be made by the steel tape or by steel and brass
wires. The conditions of use depend on the accuracy re-
quired. Let us suppose the absolute length, coefficient of
expansion, and modulus of elasticity have been accurately
determined. Any distance can then be measured in absolute
units within an accuracy of one in one million, by taking due
precautions as to temperature and mechanical conditions.
The length of the tape for city work is usually fifty feet, and
its cross section about \ inch by ^ inch. That used in New
York City is ^ inch wide by ^ inch thick. For mining, topo-
graphical, and railroad surveying a length of one hundred feet,
with a cross-section of about \ by -^^ inch, is most convenient.
For base-line measurement the length should be from three
hundred to five hundred feet, and its cross-section from two to
three one-thousandths of a square inch. For an accuracy of
one in five thousand the tape may be used in all kinds of
weather, held and stretched by hand, the horizontal position
and amount of pull estimated by the chainmen. The tempera-
ture may be estimated, or read from a thermometer carried
along for the purpose. On uneven ground, the end marks are
given by plumb-line.
For an accuracy of one in fifty thousand the mean tem-
perature of the tape should be known to the nearest degree
Fahrenheit, the slope should be determined by stretching over
stakes, or on ground whose slope is determined, and the pull
498 SUKVEYWG,
should be measured by spring balances. The work could then
be done in almost any kind of cloudy weather. For an accu-
racy of one in five hundred thousand, extreme precautions
must be taken. The mean temperature must be determined
to about one fifth of a degree F., the slope must be accurately
determined by passing the tape over points whose elevations
above a given datum are known, the pull must be known to
within a few ounces, and all friction must be eliminated. The
largest source of error is apt to be the temperature. On clear
days, the temperature of the air varies rapidly for varying
heights above the ground, and, besides, the temperature of the
tape would neither be that of the air surrounding it, nor of the
bulb of a mercurial thermometer. In fact, there is no way of
determining by mercurial thermometer, even within a few
degrees, the mean temperature of a steel tape lying in the sun,
either on or at varying heights above the ground. The work
must then be done at night or in cloudy weather^ and when air
and ground are at about the same temperature.
There should also be no appreciable wind, both on account
of its mechanical action on the tape, and from the temperature-
variations resulting therefrom.
339. Method of Mounting and Stretching the Tape.—
To eliminate all friction, the tape is suspended in hooks about
two inches long, these being hung from nails in the sides of
" line-stakes" driven with their front edges on line. These
stakes may be from twenty to one hundred feet apart. The
nails may be set on grade or not, as desired ; but if not on
grade, then each point of support must have its elevation deter-
mined. A low point should not intervene between two higher
ones, or the pull on the tape may lift it from this support.
" Marking-stakes'* are set on line with their tops about two feet
above ground, at distances apart equal to a tape-length, say 300
feet. Zinc strips about one and one half inches wide are tacked
to the tops of these stakes, and on these the tape-lengths are
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GEODETIC SURVEYING.
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marked with a steel point. These strips remain undisturbed
until all the measurements are completed, when they are
preserved for future reference. In front of the marking-stake
three " table-stakes" are driven, on which to rest the stretching
apparatus, and in the rear a " straining-stake" to which to at-
tach the rear end of the tape. These auxiliary stakes are set
two or three feet away from the marking-stake, and enough
Pig. xa9
lower to bring the tape, when stretched, to rest on the top
of the marking-stake.
The stretching apparatus is shown in Fig. 129.* A chain
is attached to the end of the tape, and this is hooked over the
* This figure, and the method here described, are taken from ihe advance*
sheets of the Report of the Missouri River Commission for 1886. The work
was in charge of Mr. O. B. Wheeler, U. S. Asst. Engr., who first used this
method on the Missouri River Survey in 1885. The author had previously
developed and used the general method, except that he stretched his tape by a
weight hung by a line passing through a loop which was kept at an angle of 45^
with the vertical, and his end marks were made on copper tacks driven into the
tops of the stakes. He had also used spring balances for stretching the tape.
So far as the author is aware, steel tapes were first used for measuring base-
lines in New Zealand in 1871, by Mr. Edwin Fairburn. The tape was 66 ft.
long, and the lengths were marked on lead. See The Surveyor (Sydney, N. S. W.)
for Sept. 21, 1900.
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500 SURVEYING,
Staple K which is attached to the block KHK. This block is
hinged on a knife-edge at H, and is weighed at K' by the load
P. The hinge bearing at H is attached to a slide which is
moved by the screw 5 working in the nut N, The whole ap-
paratus is set on the three table-stakes in front of the marking-
stake, the proper link hooked over the staple, and the block
brought to its true position by the screw. This position is
shown by the bubble L attached to the top of the block. If
the lever-arms HK and HK' are properly proportioned, the
pull on the tape is now equal to the weight P. To find this
length of the arm HK, let HK^ k ; HK' ^k\ the horizontal
distance from the knife-edge H to the centre of gravity of the
Mock =^; and the weight of block = B.
Then, taking moments about H, we have
Pk^Pk^Bgork^k'-^^pg. .... (I)
When equation (i) is fulfilled then the pull on the tape is just
equal to the weight P, when the bubble reads horizontally. The
centre of gravity of the block is found by suspending it from
two different axes and noting the intersection of plumb-lines
dropped from these axes.
At the rear end the tape is held by a slide operated by an
adjusting screw similar to that shown in Fig. 129. This slide
rests on the straining-stake, and the rear-end graduation is
made to coincide exactly with the graduation on the zinc
strip which marked the forward end of the previous tape-
length. The rear observer gives the word, and the forward end
is marked on the next zinc strip. The thermometers are then
read, and the tape carried forward.*
The measurement is duplicated by measuring again in the
same direction, the zinc strips being left undisturbed.
In obtaining a profile of the line the level rod is held on
the suspension nails and on a block, equal in height to the
length of the hooks, set on top of the marking-stakes.
♦ The U. S. C. & G. Survey now (1900) use the steel tape in measuring
primary bases, in shorter lenj^ths than here recommended. In this case the sus-
pension from hooks is found unnecessary, and the pull is given by means of
levers retting on the ground and the use of spring balances.
GEODETIC SURVEYING. lO\
For transferring the work to the ground, or to a stone set
beneath the surface, a transit is mounted at one side of the
line and the point transferred by means of the vertical motion
of the telescope, the line of sight being at right angles to the
base-line.
340. M. Jaderin's Method. — Prof. Edward Jaderin, of
Stockholm, has brought the measurement of distances by
wires and steel tapes to great perfection. He uses a tape 25
metres in length, and stretches it over tripods set in line, as
shown in Fig. 130. On the top of the tripod head is a fixed
graduation. At the rear end of the tape there is a single grad-
uation, but at the forward end a scale ten centimetres in length
Fig. 130.
is attached to the tape, this being graduated to millimetres on
a bevelled edge. The middle of this scale is 25 metres from the
graduation at the other end of the tape. The tripods are set
as near as may be to an interval of 25 metres, but it is evident
that the reading may be taken on them if this interval is not
more than 5 centimetres more or less than 25 metres. The
reading is taken to tenths of millimetres, the tenths being
estimated. The tape is stretched by two spring balances, a
very stiff spring being used at the rear end and a very sensi-
tive one at the forward end. The rear balance simply tells the
operator here when the tension is approximately right, the
measure of this tension being taken on the forward balance,
which is shown in the figure.
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502 SURVEYING.
If a single steel wire or tape be used, Mr. Jaderin also
finds that the work must be done in cloudy and calm weather,
or at night, if the best results are to be obtained. But he
finds that if two wires be used, one of steel and the other of
brass, he can continue the work during the entire day, even in
sunshine and wind, and obtain an accuracy of about one in
one million in his results.* The wires are stretched in succes-
sion over the same tripods, by the same apparatus, one wire
resting on the ground while the other is stretched. More ac-
curate results could doubtless be obtained if both wires are
kept ofl the ground constantly, the wire not in use being held
by two assistants, or if stakes and wire hooks are used, both
wires might be stretched at once in the same hooks. The two
wires form a metallic thermometer, the difference between the
readings of the same distance by the two wires determining
the temperature of both wires, when their relative lengths at a
certain temperature and their coefficients of expansion are
known. This method is similar in principle to that of the
Coast Survey apparatus, where steel and zinc bars are used,
shown in Fig. 128. In such cases the true length of line is
found by equation (5), p. 509.
At least three thermometers should be used on a 300-foot
tape, and they should be lashed to the tape or suspended by it at
such points as to have equal weight on determining its tempera-
ture. Thus if the tape is 300 feet long the thermometers should
be fastened at the 50, 150, and 250 foot marks. They should of
course have their corrections determined by comparison with
some absolute standard or with other standardized thermom-
eters.
* See " Geodiltische L^ngenmessung mit Stahlbanden und MetalldrlUiten/*
von Edv. jaderin, Stockholm (1885, 57 pp.) Also, ** £xpos6 616mentaire de la
nouvelle Methode de M. Edouard Jaderin pour la mesure des droites g6od6-
siques au moyen de Bandes d'Acier et de Fils m6talliques/' par P. E. Bergstrand,
Ing^nieur au Bureau central d'Arpeniage, k Stockholm (1885, 48 pp.). See
also U. S, C. 6* G, Survey Report for 1893, p. 125, for a complete translation ol
this report.
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GEODETIC SURVEYING. 5^3
If the appliances above outlined be used with a single tape
or wire, and the work be done on calm and densely cloudy
days, or at night, or with two wires used even in cl^r weather,
it is not difficult to make the successive measurements agree to
an accuracy of one in five hundred thousand. There still re-
mains, however, the errors in the absolute length, in the coeffi-
cient of expansion, in the modulus of elasticity, in the measure
of the pull, and in the alignment, none of which would appear
in the discrepancies between the successive measurements.
341- The Absolute Length is the most difficult to deter-
mine. The best way of finding it would be to compare it with
another tape of known length. The U. S. Coast and Geodetic
Survey now make comparisons of steel tapes up to loo feet \n
length for a small fee.*
If an absolute standard is not available, then the length may
be found by measuring a known distance, as a previously
measured base-line, and computing the temperature at which
the tape is standard. Or the tape may be compared with a
shorter standard, as a yard or metre bar, by means of a com-
parator furnished with micrometer microscopes.f
* The absolute length of the 300-foot steel tape belonging to the Mississippi
River Commission, the coefficient of expansion and the modulus of elasticity of
which the author himself determined in 1880, has now been obtained. This was
done by measuring a part of the Onley Base Line with this tape, using the
method herein outlined. This base is situated in Southern Illinois, and forma
the southern extremity of U. S. Lake Survey primary trlangulation-system. The
probable error in the length of the base, from the original measurements, was
about one one-millionth. The recent tape-measurements are remarkably accor-
dant, so the length of this tape is now very accurately known. A similar tape
belonging to the engineering outfit of Washington University has been com-
pared with this one at diflferent temperatures, and its absolute length and coeffi-
cient of expansion found. The 50-foot subdivisions have also been carefully
determined.
f Such an apparatus is used in the physical laboratory of Washington Uni-
versity, which, in conjunction with a standard metre bar which has been com-
pared with the European standards, enables absolute lengths to be determined
to the nearest one-thousandth of a millimetre.
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504 SURVEYING.
342. The Coefficient of Expansion may be taken any
where from 0.0000055 to 0.0000070 for 1° F.* If the tape is
used at nearly its standard temperature, then the coefficient of
expansion plays so small a part that its exact value is unim-
portant. If it is used at a temperature of 70° F. from its
standard temperature, and if the error in the coefficient used
be twenty per cent, the resulting error in the work would be
one in ten thousand. This is probably the extreme error that
would ever be made from not knowing the coefficient of ex-
pansion, some tabular value being used. If nothing is known
of the coefficient of expansion, probably 0.0000065 would be
the best value to use. It is evident, however, that for the
most accurate work the coefficient of expansion of the tape
used must be carefully determined.
* The author made a series of observations on a steel tape 300 feet long, the
readings being taken at short intervals for four days and three nights. The
tape was enclosed in a wooden box, and supported by hooks every sixteen
feet. The observations were taken on fine graduations made by a diamond
point, there being a single graduation at one end, but some fifty graduations a
millimetre apart at the other end. The readings were made by means of
micrometer microscopes mounted on solid posts at the two ends. The range
of temperature was about 50* F., and the resulting coefficient of expansion for
I* F. was o 00000699 ± 3 in the last place. The coefficient for the Washington
University tape is 0.00000685. Prof. T. C. Mendcnhall found from six or eigh\
experiments on steel bands used for tapes, a mean coefficient of 0.0000059.
Steel standards of length have coefficients ranging from 0.0000048 to 0.0000066.
Mr. Edward JSderin, Stockholm, has obtained a mean value of 0.0000055,
from a number of very careful determinations, both from remeasuring a primary
base-line, and from readings in a water-baih. Several steel wires were tested,
and their coefficients all came very near the mean as given above.
For brass wires he found a mean coefficient of 0.0000096 F. The isfoot
standard brass bar of the U. S. Lake Survey has a coefficient of o.ooooioo,
while tabular values are found as high as 0.0000107 F.
There is some evidence that cold-drawn wires have a less coefficient of expan-
sion than rolled bars and tapes.
Coefficients of expansion have seldom been found with great accuracy, the
coefficients of the " Mdtre des Archives," the French standard, having bad ao
erroneous value assigned to it for ninety years
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GEODETIC SURVEYING, 505
343. The Modulus of Elasticity is readily found by ap-
plying to the tape varying weights, or pulls, and observing the
stretch. The correction for sag will have to be applied for each
weight used, in case the tape is suspended from hooks, which
should be done to eliminate all friction.
Let'P, be the maximum load in pounds ;
P, " " minimum load in pounds ;
a " ** increased length of tape in inches due to the
increased pull ;
L " " length in inches for pull /*., or the graduated
length of tape ;
^ " " cross-section in square inches ;
E " ". modulus of elasticity ;
d " " distance between supports in inches ;
w " " weight of one inch of tape in pounds ;
s " " shortening effect of the sag for the length L ;
V " " sag in inches midway between supports.
Then we have
But for the pull P,, the shortening from sag is much less
than for the pull P^. We must therefore find the effect of the
sag in terms of the pull.
344. Effect of the Sag. — Where the sag is small, as it
always is in this work, the curve, although a catenary, may be
considered a parabola without an appreciable error.
If we pass a section through the tape midway between sup-
ports, and equate the moments of the external forces on one
side of this section, we obtain, taking centre of moments at
the support,
_ wd d wd^
or
^ = ^- 0)
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5c6 SCkP^LYIXG,
If the length of a parabolic curve be given by an infinite
series, and if all terms after the second be omitted, which they
V
may when , is small, then we may write —
Length of curve = ^f I +-^j (2)
If we now substitute for v its value as given in equation
(i), we have
Length of curve = rf| i -] \~p] \ •
If we call the excess in length of curve over the linear dis-
tance between supports the effect of the sag^ we have
dlwdV . ,
2^\P
for one interval between supports. If there are n such inter-
vals in one tape-length, then nd = Z, and the effect of the sag
in the entire tape-length is
^.=a(^7 (4)
If 5, and S^ be the effects of the sag for the pulls P^ and P^
(S^<iS^ for P^'>P^\ then the total movement at the free end
due to the pull being increased from P^ to /*, would be ^?-f"
(5o — 5,). If this total movement be called My then we would
have
^_ (p,-p:)l />,-/>. ,^^
S{M- S, + S,)- ^(M {wd)l {PI -P:s\
■^^L 24 \ p:p: )}
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GEODETIC SURVEVING, 507
Example.
Let A = 60 pounds;
7*0 = 10 pounds;
w = 0.00055 pound per inch of tape;
d = 300 inches = 25 feet;
S = 0.002 square inch;
M = 3.2 inches;
L = 3600 inches = 300 feet.
To find £.
From equation (5) we have
50
£ = 7 7 -r- = 28,500,000.
0.002
j 3-2 _ 0.027/ 3500 \
i 3600 24 ^36ooooy
From the same data, we find from eq. (4) the effect of the sag to be 0.040
inch for the ten-pound pull, and o.ooi inch for the sixty-pound pull.
Evidently, if the tape is stretched by the same weight when its absolute
length is found, and when used in measuring, the stretch, or elongation from
pull, would not enter in the computation, and so the modulus of elasticity
would be no function of the problem.
Again, the stretch per pound of pull may be observed for the given tape, and
then neither £ nor 5, the cross-section, would enter in the computation.
345. Temperature Correction. — If mercurial thermome-
ters are used, their field- readings must first be corrected for
the errors of their scale-reading, each thermometer having, of
course, a separate set of corrections. Then the mean of the
corrected readings may be taken for all the whole tape-lengths
in the line measured, and the correction for the entire line
obtained at once. Thus,
let L = length of line ;
7*0 = temperature at which the length of the tape is given
for the standard pull /*^, this usually being the tem-
perature at which its true length is its graduated
length for that standard pull ;
Tm = the mean corrected temperature of the entire line;
a = coefficient of expansion for 1° ;
Ct = correction for temperature.
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5CS SURVEYING.
Then C,^-{-a{T^-T,)L (i)
The temperature correction for a part of a tape-length is com-
puted separately.
If the value of a for the tape used is not known, it may be
taken at 0.0000065.
If a metallic thermometer is used, as a brass and a steel
wire, or a brass and a steel bar as in the U. S. C. and G. S.
apparatus shown on p. 494, then we have the following :
346. Temperature Correction when a Metallic Ther-
mometer is used.
Let / = length of wire or tape used, as 300 feet ;
4 = absolute length of the steel wire at the standard
temperature of, say, 32^ F. ;
4 = same for brass wire ;
L = total length of line for whole tape-lengths ( — r:^
approximately) ;
n ••= number of lengths of the standard measured ;
r, = mean value of all the scale-readings on steel wir*
for the entire line \^ = — -^ ;
Tft = same for scale-readings on brass wire ;
a^ — coefficient of expansion for the steel wire ;
a^=z " " " " " brass "
t^ = mean temperature for the entire line.
Then we have
L =: «(/. + r.) (I + {U - 32°)«'.) ) . ,
= «(4 + r»)(i+(/.-32>»))- • • • W
Since the temperature correction is relatively a very small
quantity, we may put /, + r, = 4 + ^6 = A the length of the
tape to which the temperature correction is applied.
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GEODETIC SURVEYING.
509
We then have from (2)
(3)
Substituting this value of the temperature in (2), we obtain
/. = «[/.+ r. + ^-Z^ ((4 + r.)- (4 + r»))]. . (4)
. If we put 4 -f- r, = 5, and 4 -f- r^ = 5^, we have
(5)
From either of the equations (5) we may compute the length
of the line as corrected for temperature. If, however, it is
desired to find the temperature correction separately, in order
to combine it with the other corrections, we have
AT.
C^=«(5.-5,)— -^-,
^6-«^a
(6)
for the temperature correction to be applied to the measured
length by the steel wire, or
Cu = «(5. — S^
ot^ — oc.
(;)
as the temperature correction to be applied to the measured
length by the brass wire.
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5^0 SURVEYING,
These formulae all apply only to the entire tapeUengihs, Any
fractional length would have to be computed separately, or
else a diminished weight given to their scale-readings in obtain-
ing the mean values, r^ and r^^.
347. Correction for Alignment, both horizontal and ver-
tical.— The relative elevations of the points of support are
found by a levelling instrument, and the horizontal alignment
done by a transit or by eye. An alignment by eye will be
found sufficiently exact if points be established on line by
transit every 500 or 1000 feet. The suspending nails and hooks
afford considerable latitude for lateral adjustment when the
tape is stretched taut ; hence the horizontal deviation will be
practically zero unless the stakes are very badly set, and the
relative elevations of any two successive supports should be
determined to less than 0.05 foot. If no care is taken to have
more than two suspension points on grade, then each section
of the tape will have a separate correction. Usually a single
grade may as well extend over several sections, in which case the
portion on a uniform grade may be reduced as a single section.
Let /„ /„ /„ etc., be the successive lengths of uniform grades^
and A„ A,, A„ etc., the differences of elevation between the
extremities of these uniform grades ; then for a single grade we
would have the correction
or P -2Cl-{-C = r-h\
But since C is a very small quantity as compared with /,
A"
we may drop the C, whence we have C = -^ for a single grade.
The exact value of C, in ascending powers of A, is
^=7/+87'^ i6/^ + ^^"- .... (I)
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GEODETIC SURVEYING, 511
For the entire line, if all but the first term be neglected,
the correction is
Hit: ,h: ,h: h:\
^''--2lx + x + ^+ • • • 4"/ ^^^
If the /*s are all equal, as when no two successive suspen-
sion points fall in the same grade, then we have
c; = -2/(*.'+v+>4.'+ • • . V) = --^. . (3)
Since the relative elevations are determined, and not the
angles of the grades, these formulae are more readily applied
than one involving the grade angles. '
The error made in rejecting the second power of C in the
above equations is given in the table on the following page,
where / and h are taken in the same unit of length.*
If the grades are given in vertical angles, as they always
are with the ordinary base apparatus, then we have for the
correction to each section whose length is /, and whose grade
is a above or below the horizon,
e
Cg = — /(i — COS ^ = — 2/ sin* -.
If 6 be expressed in minutes of arc, and if the grade angle
is less than about six degrees, or if the slope is less than one in
ten, we may write
r = _ 2/sin» - = _ i/^ sin" I' = - ^^^ (fl
'22 2
= —0.00000004231 ^/;
* From jaderin's GeodStische Lftngenmessung.
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512
SURVEYING,
or by logarithms,
log Cg = const, log 2.626422 + 2 log 0 + log /.
A*
TABLE OF ABSOLUTE ERRORS IN THE FORMULA Q=^^
Length of
Uniform
Grade.
Absolute Error in the Same Units used for /and k.
0.00005
0.00015 0.00025 0.00035
A = Rise or Fall in Length /.
0.00045
I
2
3
4
5
0.14
.24
.32
.40
.47
0.19
.31
.42
.53
.62
6
7
8
9
10
.54
.61
.67
.73
.79
.71
.80
•88
.97
1.05
0.81 1
.01 1
1. 00
1. 10 1 1.19
I . 19 1 I . 29
II
12
13
14
15
.85
.91
.97
1.02
1.08
1. 12
1.20
1.27
1-35
1.42
1.28 1.39
1.36 1.48
1.45 1.57
1.53 1.66
I. 61 1.75
1.67
1.77
1.86
16
17
18
19
20
1. 13
1. 18
1.24
1.29
1.34
1.49
1.56
1.62
1.69
1.76
1.69 1.84
1.77 1.92
1.85 2.01
1.92 2.09
2 00 ! 2.17
1.96
2 05
2.14
2.23 1
2.31
21
22
23
24
25
1.39
1.44
1.48
1.53
1.58
1.82
1. 89
1.95
2.02
2.08
2.07 2.25
2.15 2.33
2.22 2.41
2 . 29 2 . 49
2 36 1 2.57
2.40
2.48
2.57
2.65
2.73
26
27
28
29
30
1.63
1.67
1.72
I 77
1. 81
2.14
2.20
2.26
2.32
2.38
2.43 2.65
2.50 1 2.72
2.57 2.80
2.64 2.87
2.71 1 2.95
2.82
2.90
2.98
3.06
3.14
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GEODETIC SURVEYING, $13
348. Correction for Sag. — From equation (4), p. 506, we
have
^■=-~^ (4)
If the standard length be given with the pull P^, and the
distance between supports d^y while in the field the pull P and
distance d between supports be used, then the correction for
sag is
where Z, d, and C, are taken in the same unit of length, and w
is the weight of a unit's length of tape in the same units used
ioxP.
349. Correction for Pull. — From equation (i), p. 505, we
may write at once
^^ " "^ SE '
Here P is taken in pounds. L and Cp in inches, and 5 in
square inches, since E is usually given in inch-pound units. If
E has not been determined by experiment, it may be taken at
28000000. The cross-section 5 is best found by weighing the
tape and computing its volume, counting 3.6 cubic inches to
the pound. Knowing the length, the cross-section can then be
found. If the stretch has been observed for different weights,
and the value of E computed, the value of 5 is of no conse-
quence, provided the same value be used for both observations.
350. Elimination of Corrections for Sag and Pull.—
Since the correction for sag is negative and that for pull is
positive, we may make them numerically equal, and so elimi-
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5H SURVEYING,
nate them both from the work. If this be done, the normal
or standard length of the tape should be obtained for no sag
and no pull, and its normal or standard temperature found such
that at this temperature, and for no sag and no pull, its gradu-
ated length is its true length.
i \i T^ is the temperature at which the tape is of standard
length for the pull P^ and the distance d^ between supports,
and if / is the length of the tape, then we have,
Shortening from sag = - (—5-) ,
PI
Lengthening from pull = -^,
or net lengthenmg from sag and pull = cZ- — r: VP') '
Lengthening from x degrees F. = xal.
If, therefore, the effects of sag and pull were eliminated,
the tape would be of standard length at a temperature x
degrees above T^j where
^-m-km <■)
where all dimensions are in inches and weights in pounds.
The standard temperature for no sag and no pull would be,
therefore,
7;=7; + ;r. (2)
We will call this the normal temperature.
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GEODETIC SURVEYING. S'S
In order that the corrections for sag and pull shall balance
each other, we must have
SE~24\P)'
' /SE~
or ^• = \/-2i:^""'^'' ^3)
which we will call the normal tension.
If the stretch in inches is known for one pound of pull for
the given tape, we may call this ^, and we will have
Also, /w = W= weight of entire tape between end graduations,
W
or w = —J-'
I
And ^ = « = number of sags in the tape.
Substituting these values in (3), we obtain
^-=y^^r (4)
where W^ weight of entire tape in pounds;
/ = length of tape in inches ;
e = elongation of tape for a one-pound pull ;
n = number of sags in tape = ^.
If the tape has no intermediate supports, then « = i, and
we have for the normal tension
= V—
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5l6 SURVEYING,
Example. — For the 3oo-foot steel tape, whose constants the author deter-
mined, we have fV = 2 lbs., /= 3600 inches, e = 0.066 inch. If the supports
are 30 feet apart, n =. 10, whence, from eq. (4). Pn = 4.48 pounds.
If ;f = 6, or if the supports were placed 50 feet apart, we would find Pn =
6.32 pounds.
If » = 3, or if the supports are 100 feet apart, Pn = 10.03 pounds.
In the last case, the sag would be ten inches midway between supports.
351. To reduce a Broken Base to a Straight Line.—
It is sometimes necessary or convenient to introduce one or
more angles into a baseline. These would never deviate much
from 180°. Let the difference between the angle and 180° be
6, and let the two measured sides be a and b, to find the side c.
If 6 be expressed in minutes of arc and if it is not more than
about 3^, the following approximate formula will prove suf-
ficiently exact :
. , , , sin* I' abd"
side ^ = a + *
2 ' a + b
= a + ^ — 0.00000004231 , y
If 6 is greater than from 3^ to 5*^, the triangle would have to
be computed by the ordinary sine formula.
352. To reduce the Length of the Base to Sea-level.
—In geodetic work, all distances are reduced to what they
would be if the same lines were projected upon a sea-level
surface by radii passing through the extremities of the lines.
It is not necessary, however, to reduce all the lines of a trian-
gulation system in this manner, since if the length of the base-
line is so reduced the computed lengths of all the other lines
of the system will be their lengths at sea-level. The angles
that are measured are the horizontal dLVigl^s, and are not affected
by the differences of elevation of the various stations. It is
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GEODETIC SUKVEYING, 51/
necessary, therefore, to know the approximate elevation of the
base above sea-level.
Let r — mean radius of earth ;
a = elevation above sea-level ;
£ = length of measured base ;
d = length of base at sea-level.
Then r + a \r\\B\b,
or b^ B ^
r + a
The correction to the measured length is always negativei
and is
c=*-5=-5(i-4-) = -5(4-).
Since a is very small as compared to r, we may write
The mean radius* in feet is
20026062 + 20855121
mean r = — ^^ ^ ^ = 20890592 feet,
log r (in feet) = 7-3199507.
353. Summary of Corrections.— For the significance of
the notation used in the following equations, see the preceding
articles where they are derived. The corrections are all for
* Rigidly, we should use the length of the normal for the given latitude, but
the mean radius as above found is sufficient for most cases.
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5l8 SURVEYIXG.
the entire line measured, or rather for that portion of it com-
posed of entire tape-lengths, and are to be applied with the
signs given to the measured length.
I. Correction for Temperature.
For a single standard with mercurial temperatures,
c; = + a(r„ - 7;)z (o
For metallic thermometer-readings, as found from steel and
brass standards, for instance, the correction to be applied to
the length as found by the steel wire, or standard, is
G = «(5.-S.)^-. (2)
2. Correction for Grade.
In terms of the difference of elevation of grade, points at a
common distance, /, apart,
%yg — 2/'* * * ^* * * * ^^^
In terms of the grade angles, expressed in minutes of arc
CJ, = — 0.000000042312'^/. (4)
3, Correction for Sag.
For the standard length given for a pull P., and a distance
between supports ^/„ while P and d are used in the field-work,
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GEODETIC SURVEYING. 510
- - — -----
For the standard length gfiven for no pull and no sag,
c. = -i^- (^
4. Correction for PulL
^-= + ^:^*^' (7)
or C;=(/>-/>.y«. (8)
5. To reduce Standard Temperature to Normal Temperature,
When the temperature of the tape (T^ is known at which
the graduated is the true length for the pull P^ and distance
between supports ^/„ to find the corresponding temperature for
no pull and no sag, this being called the normal temperature
(Tn), we have, in degrees,
^-=^.+iS-i(^=)'] (»)
6. To eliminate Corrections for Sag and Pull.
Make the pull /', = a/|^(w</)*; (lo)
y 2^n*
or P^'^K TJT:,^ (")
For no intermediate supports to tape,
'-7S (-)
24^
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520 SURVEYING,
P^ IS called the normal tension.
7. Correction for Broken Base,
If a and b are the two measured sides which make an angle
of 180° — ^, the correction to be added to a -{- b\,o get the
distance between their extremities, Q being less than 5°, and
expressed in minutes of arc, is
abS'
G = — 0.00000004231 ^^-17^.
8. Correction to Sea-level
r
where L is the length of the measured base at an altitude a
above sea-level.
log r (in feet) = 7.3199507.
354. To compute any Portion of a Straight Base which
cannot be directly measured. — It sometimes is convenient
to take a base-line across a stream or other obstruction to di-
rect measurement. In such a case a station may be chosen
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GEODETIC SURVEYING. 521
as O in Fig. 131, and the horizontal angles AOB = P, BOC =
Q, and COD =zR measured. If the parts AB and CD lie in
the same straight line, and AB = a and CD = b are known,
then BC = x may be found by measuring only the angles at O.
Thus in the triangles ABO and ACO we have
CO x + a smP
B0~~ a sin(P+0'
also from the triangles BDO and CDO we have
CO b sin {Q + R) /
BO'^ X -^b sin ie '
Let Ar=:/*+ (2 and Z= G + ^, then by equating the
above values of -^ we have
( \ \( X ix ^* (sin A' sinZ)
whence
a^b
/^^(sin A^sinZ) f^^V
Y sinPsinie +\ 2 /'
Evidently only the positive result is to be taken.
The points A, O, and D should be chosen so as to give
good intersections at A and D,
355. Accuracy attainable by Steel-tape and Metallic-
wire Measurements. — The following results have been at-,
tained by using the methods herein described :
I. In Sweden, Mr. Edw. Jaderin measured a primary
base-line two kilometres in length three times, by means of
steel and brass wires 25 metres long, in ordinary summer
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522 SURVEYING.
weather, mostly clear, with a probable error of a single deter-
mination of I in 600,000, and a probable error of the mean re-
sult of I in 1,000,000, as compared with the true length of the
line as obtained by a regular primary base apparatus.*
2. On the trigonometrical survey of the Missouri River,
in 1885, Mr. O. B. Wheeler, U. S. Asst. Engineer, obtained
the following results, using one steel-tape 300 feet long:
Glasgow Base,
First measurement 7923.237 feet.
Second " 7923.403 "
Mean 7923.320 ± 0.056 feet.
In this case the sun was shining more or less on both
measurements. The probable error of a single result is i in
100,000, and of the mean of two measurements I in 140,000.
Benton Base.
First measurement 9870.443 feet.
Second " 9870.388 **
Mean 9870.415 ± O.018 feet.
The probable error of a single measurement is i in 380,000,
and of the mean, i in 533,000.
Trovers Point Base,
First measurement 971 1.915 feet.
Second " 971 1.892 **
Mean 971 1.904 ± o 0078 feet.
* For ilile of Mr. Jsiderin's pamphlet describing his methods and results see
foot-note, p. 502.
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GEODETIC SURVEYING, 523
The probable error of a single measurement is i in <jpo,OO0y
and of the mean it is i in 1,250,000.
Olney Base,
First measurement 1082 1.9658 feet.
Second " 10821.9665 "
Mean 10821.9662 ± 0.0002 feet.
This base had been measured by the U. S. Lake Survey
Repsold base apparatus, with a probable error of about i in
1,000,000. This portion of it, about half the entire base, was
remeasured with the tape in order to determine the absolute
length of the tape. The work was done on both the tape-
measurements in a drizzling rain, so that the temperatures
were obtained with great accuracy. The mean tempera-
tures of the two measurements diflfered, however, by several
degrees, so that the two sets of graduations on the zinc strips
were quite divergent, and it was only after the final reduc-
tion that the two results were known to be so nearly identical.*
3. The author has measured a number of bases about one
half mile in length, in connection with students* practice sur-
veys, by the methods given above, and in each case obtained a
probable error of the mean of three or four measurements of
less than one-millionth part of the length of the line. The
work was always done on densely cloudy days, all the con-
stants of tape and thermometers being well determined.
Note. — Prof. R. S. Woodward when assistant on the U. S. C. and G. Survey,
in 1892, made five measurements of a base line 3,807 metres long, in four sections,
using two steel tapes, making two measurements with each at night, and one
measurement in the daytime in clear sunlight. These results gave a probable
error in the mean of all of the results of part, not including the error in
2,000,000
the length of the tape itself, and a probable error of o'koooQ ^^^ ^^^" ^^^
sources of error arc taken into account. See a paper on The Use of Long Steel
Tapes for Measuring Base Lines, Tr,ins. Am. SiC. C E., Vol. XXX. (1893), p. 81,
m '
* From the Report of the Missouri River Commission, i880-
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Fia X39.
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GEODETIC SURVEYING, 525
MEASUREMENT OF THE ANGLES.
356. The Instruments used in triangulation are designed
especially for the accurate measurement of horizontal angles.
This demands very accurate centring and fitting at the axis, and
strict uniformity of graduation. It was formerly supposed that
the larger the circle the more accurate the work which could
be done. It is now known that there is no advantage in having
the horizontal limb more than ten or twelve inches in diameter.
There are two general methods of reading fractional parts
of the angle, smaller than the smallest graduated space on the
limb. One is by verniers, the other by micrometer micro-
scopes. Verniers may be successfully used to read angles to
the nearest ten or twenty seconds of arc, but if a nearer ap-
proximation is desired microscopes should be employed.
Fig. 132 shows a high grade of vernier transit, capable also
of reading vertical angles to 70°. Its horizontal limb is 8
inches in diameter and reads by verniers to ten seconds. It
may be used as a repeating * instrument, and used either with
or without a tripod. To mount such an instrument upon a
station or post, a trivet, made of brass and shown in Fig. 135, is
used. The pointed steel legs are driven into the station, the
centre of the opening being over the station point. The arms
have angular grooves cut in their upper surface. On this trivet
may be set any three-legged instrument, so long as the radius
of its base is not greater than the length of the trivet arms.
Fig. 133 shows one of the latest forms of instruments for
reading horizontal angles in primary triangulation used on the
U. S. C. & G. Survey. It reads by three micrometer micro-
scopes to single seconds, there being an auxiliary microscope
of low power used for setting on different parts of the limb.
There is also a micrometer attachment to the eyepiece for
astronomical work. It has a twelve-inch horizontal circle and
two small vertical circles with verniers for setting approximate
* Sec Art. 359 for an explanation of this term.
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526
SURVEYING.
Fig. X33.
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GEODETIC SURVEYING.
527
Fig. 134.
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S28 sVRVEVim,
altitudes for stellar observations for azimuth. The circular
rim at bottom is for handling only. The frame of this instru-
ment, including the microscope arms, is very strong and rigid.
In Fig. 134 is shown an altazimuth instrument, or an in-
strument designed for accurately measuring altitudes as well as
the azimuths of points or lines. Both horizontal and vertical
limbs are read by means of micrometer microscopes. Such an
instrument is designed especially for astronomical observations
for latitude and azimuth, but may also be used as a meridian
or transit instrument for observ-
ing time as well as for measuring
horizontal and vertical angles in
triangulation. It is in fact the
universal geodetic instrument,
just as the complete engineer's
transit is the universal instrument
in ordinary surveying. In almost
all cases where micrometers are
Fig. 135. , . ,. , , ,
used in reading the angles the
limbs are graduated to five or ten minutes and the readings
made to single seconds.
357. The Filar Micrometer* is used for the accurate meas-
urement of small distances or angles, when the required exact-
ness is greater than can be obtained by means of a vernier
scale. It is usually combined with a microscope, the microme-
ter threads and scale lying in the plane of the image produced
by the objective. This image is always larger than the object
itself in microscopes, and therefore a given movement of the
wires in the micrometer corresponds to a very much less dis-
tance on the object sighted at, according to the magnifying
power of the objective.
* From Ji/um, thread; micros, small, and metros, measure. The thread is in
this case a spider's web, or scratches on glass.
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CnODETIC SURVEYING.
529
The frame holding the movable wires has a screw with a
very fine thread working in it, called the micrometer screw.
This screw has a graduated cylindrical head, or disk, attached
to it, there usually being sixty divisions in the circumference
when used in angular measurements. The number of whole
revolutions are recorded by noting how many teeth of a comb-
scale are passed over, this scale being nearly in the plane of the
wires and therefore in the focus of the eye-piece. The frac-
tional parts of a revolution are read on the graduated screw-
head outside. These micrometer attachments are shown on
the two microscopes in Fig. 133 and on the five in Fig. 134.
Fig. 136.
Fig, 136 is a sectional view of a filar micrometer. The graduat-
ed head h is attached to the milled head m, forming a nut into
which the micrometer-screw a works. This screw is rigidly at-
tached to the frame b, to which are fastened the movable wires
f. The comb-scale s and fixed wire / are attached to the
frame Cy which is adjusted to a zero-reading of the graduated
head by the capstan-screw d. The lost motion on both of
these frames is taken up by springs. The complete revolutions
of the screw are counted on the comb-scale, and the fractional
part of a revolution on the graduated head. The reading is
made by bringing the double wires symmetrically over a grad-
uation, the space between the wires being a little more than
the width of the graduation, when the exact number of revolu-
tions and sixtieths are read on the comb-scale and on the head.
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SjO ' . . SURVEYING. ' "^'-c"
If the limb is graduated to ten minutes and each revolution
corresponds to one minute, then if the reading is taken on the
nearest^ graduation, the number of revolutions need never ex-
ceed five. If, however, the reading be always taken to the last
ten-minute mark counted on the limb, then ten revolutions raay
have to be read on the screw. The movement of the threads
is as they appear to be, there being no inversion of image be-
tween wires and eye. The movement on the limb is, however,
opposite from the apparent motion.
If the limb is graduated to ten minutes, and a single revo-
lution of the screw corresponds to the space of one minute,
thenjust ten revolutions of the screw should move the wires
fronrx one graduationto the next. If this is not exactly true,
then the value of a ten-minute space should be measured a
number of times, by running the wires back and forth, the
mean result taken, and from tliis the value of one revolution of
the screw determined. This value ,\s called the *f run of the
screw," and a col*rection.i$- applied to the readings, which are
always made in degrees, -rfiiputcs, and seconds, counting one
revolution a minute and one divisfon on the head a second of
arc. This correction is called "correction for run,** and should
be determined for all parts of the screw used. If the value of
one revolution is not exactly wtrat it is designed to be, it can
be adjusted by moving the objective of the microscope in or
out a little, or the whole microscope up or down with refer-
ence to the limb, thereby changing the size of the image.
.Even wlien tliis adjustment is accurately made, there may be
still a correction for run on account of the screw-threads not
being of uniform value. In this case the value of each revolu-
tion of the scre\V is deterniined independently, these values
tabulated^ and the correction for run from this source deter-
mined for any given reading. Again, as the microscope re-
volves around the limb with the alidade, the plane of the
graduations may not remain at a constant distance from the
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^ GEODETIC SURVEYING, i Sl^
objective, in which case the size of the image would vary to a
corresponding degree. To determine this, the values of ten-
minute spaces are determined on various parts of the Jimb|
and if these are not constant, then a table of corrections forrui)
may be made out for different parts of the circle.*.
For reading on graduated straight lines the double threads
give better results than either the single thread or. the inter-
secting threads. The space between the threads should be a
little greater than the. width of the image of the graduation-
line, so that a narrow strip of the limb's. illuminated upper
surface may appear oa either side of the graduation and inside
the wires. The setting Js then made so as to make these iili^-
minated lines of equal width. It is conceded that such an ar-
rangement will give more exact readings than any other that
has been used.
The magnifying power of the microscope is from thirty tp
fifty.
358. Programme of Observations. — There are two gen-
eral methods of reading angles in triangulation work. One
method consists in measuring each angle inde-
pendently, usually by repeating it a number of
times by successive additions on the limb, and
then reading this multiplied angle, which is di-
vided by the number of repetitions to give the
true value of the angle. In the other method
the readings are made on the several stations in
order, as A, B, C, A and E, in the figure, and
the angles found by taking the difference between
the successive readings. Each method has its
advantages and disadvantages. If the instrument has an ac-
curate fitting in the axis, clamps which can be set and loosened
without disturbing the positions of the plates, is provided with
verniers which have a coarse reading, as twenty or thirty sec-
onds, and accurate work is desired^ and if such an instrument
* On ihe U. S. C. & G. Survey the correction for run is eliminated by chang-
ing the setting on the initial station Ly 5 min. -v- n each lime, where n is t
number of sets of readings taken. This gives readings over all parts of the
of the screw.
53^ SURVEYING.
is mounted on a low, firm station, then the method by repeti-
tion would give superior results. If any of these conditions are
not fulfilled, and especially if the instrument is provided with
micrometer microscopes, whereby readings may be taken to
the nearest second of arc, it is much more convenient, cheaper,
and generally more accurate to read the stations continuously
around the horizon, back and forth, until a sufficient number
of readings have been obtained.
359, The Repeating Method. — This method was for-
merly used almost exclusively, but the other is the only one
^ now used with the most accurate instruments.'*^ It was found
^"t ^" that systematic errors were introduced in the method by
^ repetition of a single angle, due largely to the clamping appa-
ratus. If this method is used the repetitions should be made
\ first towards the right and then towards the left ; the number
of repetitions making a set should be such as to make the mul-
tiplied angle a multiple of 360°, as nearly as possible, so as to
eliminate errors of graduation on the limb. Thus, for an angle
of 60° repeat it six times and then read. For the second set
repeat six times in the opposite direction, and with telescope
inverted. If triangulation work is to be done with the ordi-
dary engineer's transit, which reads only to 30 seconds or one
minute, this method may give very fair results, provided there
is no movement of circles from the use of the clamping apparatus
and no lost motion in the axes. The programme would be as
follows: . "''((. . , ^ Ltc^ • ^
: PROGRAMME, i , , ■ .. r-
Telescope Normal.
1. Set on left station, and read both verniers.
2. Unclamp above and set on right station.
3. " below " " left
4. " above " " right "
5. " below ** *** left "
6. " above " " right «
etc., etc.,
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GEODETIC SORVEYma, 533
until the entire circle has been traversed, then read both ver-
niers while pointing to right station. The total angle divided
by the number of repetitions is the measure of the angle
sought.
Telescope Reversed,
1. Set on right station, and read both verniers.
2. Unclamp above and set on left station.
3. " below " '* right "
4. " above " " left "
5. " below " " right "
6. " above " " left " x
until the- entife-etf€4e-ha»^beeft-tra versed by each-vermer, when
both verniers are read on the left station.
The repetition in opposite directions is designed to elimi-
nate errors from clamp and axis movements, and the revers-
ing of the telescope is designed to eliminate errors arising
from the line of sight not being perpendicular to the horizon-
tal axis, and from the horizontal axis not being perpendicular
to the vertical axis of the instrument."*^
As many such sets of readings may be made as desired,
but there should always be an even number, or as many of one-
kind as of the other. It will be observed that two pointings
are taken for each measurement of the angle, but compara-
tively few readings are made.
360; Method by Continuous Reading around the Hori-
zon.— By this method the limb is clamped in any position, and
* In case the instrument used is a theodolite, and its telescope cannot be
revolved on its horizontal axis, it should be lifted from the pivot bearings and
turned over end for end, leaving the pivots in their former bearings. If this
cannot be done conveniently, then the limb should be shifted by 360 -f- n (see
next page) each time, and this will result in mostly eliminating ihese same errors
of colliroation and inclination of horizontal axis. If it be found that the vertical
axes are not parallel, then at least four sets of readings should be taken and
these should be distributed upon the horizontal limb symmetrically with reference
to the plane of greatest inclination between the two vertical axes.
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534 SURVEYING,
left undisturbed except between the different sets of readings.
The pointings are made to the stations in succession around
the horizon, and both verniers, or microscopes, read for each
pointing. Thus, if the instrument were at o, Fig. 137, the
pointings would be made toA,Bj C, D, and E, If the telescope
is now carried around to the right until the line of sight again
falls on A, and a reading taken, the observer is said to close
the horizon : that is, he has moved the telescope continuously
around in one direction to the point of beginning. If the two
readings here do not agree, the error is distributed among the
angles in proportion to their number, irrespective of their size.
It is questionable whether such an adjustment adds much to
the accuracy of the angle values, and therefore it is common
to read to the several stations back and forth without closing
the horizon. Sum-angles can afterwards be read if desired.
Thus, after the regular readings have been taken on the sta-
tions, the angle AOE, or AOC^vlA COE, may be read, and so
one or more equations of condition obtained.
If the station is tall, there is always a twisting of its top in
clear weather in the direction of the sun's movement. This
twisting effect has been observed to be as much as i" in a
minute of time on a seventy-five-foot station. To eliminate
this action the readings are taken both to the right and to the
left. The reading of opposite verniers, or microscopes, elimi-
nates errors of eccentricity, the inverting of the telescope elimi-
nates errors of adjustment in the line of coUimation and hori-
zontal axis, and to eliminate periodic errors of graduation each
angle is read on symmetrically distributed portions of the limb.
To accomplish this the limb is shifted after each set of read-
180**
ings an amount equal to * where n is the number of sets
of readings to be taken. The following is the
^______^ ^— ^
^ For exception, see foot-note on previous page.
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GEODETIC SURVEYING,
535
1ST Set.
PROGRAMME.
Telescope normaL
Read to right.
Read to left.
Telescope inverted.
Read to right.
Read to left.
Shift the Limb.
2D Set.
Telescope inverted.
Read to right.
Read to left.
Telescope normaL
Read to right.
Read to left.
Shift the Limb.
Evidently each set is complete in itself, and as many com-
plete sets may be taken as desired, but no partial sets should
be used. If the work is interrupted in the midst of one set of
readings, the partial set of readings should be rejected, and
when the work is resumed another set begun. In reducing the
work, if one reading of any angle is so erroneous as to have to
be rejected this should vitiate that entire set of readings of
that angle.
If preferred, the telescope may be inverted between the
right and left readings, and then two readings on each mark
would constitute a complete set, when the limb could be
shifted again. If this were done, the readings at o, Fig. 137,
would be :
1ST Set \ T^^^sc^P^ Normal — Read ABCDE,
'\ " Inverted ** EDCBA,
Shift the Limb.
Set \ ^^l^s^^P^ Inverted — Read ABCDE.
\ " Normal ** EDCBA.
Shift the Limb.
361. Atmospheric Conditions.— In clear weather not even
fair results can be obtained during the greater part of the day.
From sunrise till about four o'clock in the afternoon in sum-
mer the air is so unsteady from the heated air-currents that
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53^ SURVEY 2 NG.
any distant target is either invisible or else its image is so un-
steady as to make a pointing to it very uncertain. From
about four o'clock till dark in clear weather, and all day in
densely cloudy weather with clear air, good work can be done.
If heliotropes are used, the work is limited to clear weather.
It has often been proposed to do such work at night, but the
lack of a simple and efficient light of sufficient strength has
usually prevented. The higher the line of sight above the
ground the less it is affected by atmospheric disturbances.
362. Geodetic Night Signals.— Mr. C. O. Boutelle, of the
U. S. Coast and Geodetic Survey, made a series of experiments
in 1879 ^^ Sugar Loaf Mountain, Maryland, for the purpose of
testing the efficiency of certain night signals and the compara-
tive values of day and night work. His report is given in Ap-
pendix No. 8 of the Report of the U. S. C. and G. Survey for
1880. It seems that either the common Argand or the " Elec-
tric" coal-oil lamp, assisted by a parabolic reflector or by a
large lens, gives a light visible for over forty miles. His con-
clusions are :
1. That night observations are a little more accurate than
those by day, but the difference is slight.
2. That the cost of the apparatus is less than that of good
heliotropes.
3. That the apparatus can be manipulated by the same class
of men as those ordinarily employed as heliotropers.
4. That the average time of observing in clear weather may
be more than doubled by observing at night, and thus the time
of occupation of a station proportionately shortened ; "clear-
cloudy'* weather, when heliotropes cannot show, can be utilized
at night.
363. Reduction to the Centre.— It sometimes happens
that the instrument cannot be set directly over the geodetic
point, as when a tower or steeple is used for such point. In
this case two angles of each of the triangles meeting hcrc may
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GEODETIC SURVEYING,
537
OJtl
be measured and the third taken to be i8o° minus their sum,
or the instrument may be mounted near to the geodetic point
and all the angles at this station measured from this position.
These angles will then be very nearly the same as though
measured from the true position, and may readily be reduced
to what they would have been if the true station point had
been occupied. Thus in Fig. 138 let C be the true station to
which pointings were taken from other stations, and C the posi-
tion of the instrument for measuring the angles at this station.
The Hne AB is a side of the system whose
length has been found. From the measured
angles at A and B the approximate value of
the angle C is found and the lengths of the
sides a and^ computed. At C the angle
AC'B is measured with the same exactness
as though it were the angle C itself and the
angle CC'B = a is measured by a single ob-
servation. The distance CC = r is also
found. Since the exterior angle at the inter-
section/?, RsADBf is equal to the sum of the opposite interior
angles, we have
C-+^=C"+;r,
or
C=C + {x-j^).
(0
In the triangle ACC we have the sides b and r and the
angle /4C'C known, whence
similarly
sm X =
sm^
rsin iC+a)
b
r sin a
• • f • • \2)
Since x and^ are very small angles, their sines are propor-
tional to their arcs, and we may write sin ;r = ;r sin i'' where
33
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538
SURVEYING,
X is expressed in seconds ; similarly %\x\ y ^=. y sin i'\ and equa-
tions (2) become
r sin (C' + ot)
X =
y^
b sin i'
r sin a
(3)
^sm \"
Substituting these values in (i) we have
where the correction to C is given in seconds of arc. The
signs of the trigonometrical
functions of the angle a must
be carefully attended to, as it is
measured continuously from B
around to the left to 360*".
The following is another so-
lution of the same problem :
Measure the perpendiculars
from C upon AC and BC\ Fig.
139, calling them m and n re-
's spectively. Then from equation
(l) above we have
Fio. ,39 F,c. ,40. c = C' + (;r - y\
But since the angles x and y are very small, their sines are
equal to their arcs, and we have, in seconds of arc.
x =
m
b sin i"
and
^ = :;
n
a^vci I
whence ^ = C + -v^-U, f-J - -).
' sin 1 ' V ^ al
(5)
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GEODETIC SUkVEYING.
539
There are four cases corresponding to the four positions of
C, as shown in Fig. 140. For these several cases we liave
C*-C' ^-^{~ -V 1
*' - ^» sini"U aJ'
(6)
ADJUSTMENT OF THE MEASURED ANGLES.
364. Equations of Conditions. — When any continuous
quantity, as an angle or a line, is measured, the observed value
is always affected by certain small errors. Indeed, it would
not be possible even to express exactly the value of a contin-
uous quantity in terms of any unit, as degrees or feet and
fractional parts of the same, even though this value could be
exactly determined. If, therefore, the measured values of the
three angles of a triangle be added together, the sum will not
be exactly i8o^. But we know that a rigid condition of all tri-
angles is that the sum of the three angles is i8o°. An equation
which expresses a relation between any number of observed
quantities which of geometrical necessity must exist is called
an equation of condition, or a condition equation. Thus, in
the above case, if A'y B\ and C be the mean observed values
of the angles, and A^ B, and C their true values, we would
have for our condition equation
-4 + ^+C=i8o^
(I)
♦ Log sio i" = 4.6855749.
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540
^ukvevwc.
We would also have
where a\ b\ and d are small corrections to the measured
values A\ B\ and C which are to be
determined.
Let us suppose that the length of
the side b has been exactly meas-
ured,* then when the true values of
the angles are found we may com-
pute the other two sides. If the sides
b and c have both been measured, the
length of the side c as computed from b must agree with its
measured length, and so we might write the condition equation
c =
b sin (r + O
sin(5' + d')'
(2)
Again, if the side a had been measured and its exact length
found, we would obtain the third condition equation,
a =
b sin {A' + a!)
sin {B' + b') '
(3)
We now have three independent equations involving three
unknown quantities, and can, therefore, find the quantities a\
b\ and c' , But if only one side had been measured, we should
have had but one equation from which to determine three un-
known quantities. Evidently there is an infinite number of
* This assumption is made in regard to the measured base-lines in a trian-
gulation system, since its exactness is so much greater than can be obtained
in the angle-measurements.
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GEODETIC SURVEYING, 54 1
sets of values of a\b\ and ^, which would satisfy this equation.
If we now impose the condition that the corrections shall be
the most probable ones, then there is but one set of values that
can be taken.
Equation (i) is called an an^e equation^ since only angles
are involved ; while equations (2) and (3) are called side equa-
tions^ since the lengths of the sides are also involved.
365. Adjustment of a Triangle. — The finding and ap-
plying of the most probable corrections to the measured values
of the angles of a system of triangulation is called adjusting
the system. In the case of a single triangle with one known
side and three measured angles, we have seen that there is but
one equation of condition. If the three angles have been
equally well observed, then it is most probable* that they are
all equally in error, and hence this condition of highest proba-
bihty gives us the probability equation
a'^b'^c' (4)
which enables the corrections to be determined.
Thus,let ^' + 5' + C'- i8o° = rt' + *' + ^r' = r,
then from (4) we have
«' = *'=^'=f-. (5)
where Fis the error of closure of the triangle.
* That is, this relation is more probable than are any other single relation
that can be assigned, but of course it is not more probable than all other cases
combined.
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5^2 SURVEYING,
ADJUSTMENT OF A QUADRILATERAL.
366. The Geometrical Conditions. — In the quadrilateral
in Fig. 142 there are eight observed angles, -4,, B^, B^, C„ etc.
The geometrical conditions which must here be fulfilled are :
{a) The sum of all the angles of any triangle must be 180°
plus the spherical excess* and the opposite angles at the
intersection of the diagonals must be equal.
{b) The computed length of any side, as DC^ when obtained
from any other side, as AB, through two independent sets of
triangles, as ABC, BDC, and ABD, ADC, shall be the same in
both cases.
The probability condition is that the set of corrections ap-
plied to the several angles shall be more probable than any
other one of the infinite number of sets of corrections which
would satisfy the other condition.
The condition given in {a) gives rise to the angle equations,
and that given in (d) gives one side equation.
There are evidently eight unknown corrections to be de-
termined.
367. The Angle-equation Adjustment. — In the quadri-
lateral ABCD we have four triangles in which all the angles
have been observed, two sets of opposite angles where the
other two angles of the corresponding triangles have been ob-
served, and the quadrilateral itself in which all the angles have
*It is not necessary to take account of the spherical excess in computing a
single triangle or quadrilateral ; but if azimuth is to be carried over a series of
triangles it is necessary that all the angles be spherical angles. In this place
spherical excess will be omitted ; but if it is desirable to introduce it, it is in-
serted in equations (i), (2), and (3), the right members then becoming A -}- 'i. 1%
+ ^«, and 1% + ^a, where ex is the residual excess of the angle A OB over that of
the angle DOC (being negative in this case), e% is the excess of angle BOC over
that of the angle AOD^ and /> is the spherical excess for the entire quadri-
lateral. The spherical excess may be taken as i' for each 75 square miles of
area, and this is to be divided equally amongst the angles of the figure. The
206000 y^
formula for spherical excess is ^ (in seconds) = -^ — , where A is area in
square miles, and r is radius of the earth in miles.
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GEODETIC SURVEYING. 543
been observed ; making, in all, seven geometric conditions to
be fulfilled. Only three of these conditions are independent,
however, since where any three independent conditions are
fulfilled the remaining four are fulfilled also. Thus, a great
variety of conditioned equations could be formed, but we will
.>
sNX
choose the three which give the simplest equations, viz. : that
the opposite central angles shall be equal, and that the sum of
all the angles of the quadrilateral shall be 360°. These give
rise to the following equations :
If /4„ -ff,. -ff„ C„ etc., be the observed angles, and /„ /„ and /,
the residuals in the several combinations, due to erroneous
determinations, then we have :
,8o°-M,+5.) - 1 180 - (C. + Z>.){ = /.»
or -A-B, +C + A =A. (I)
Similarly -B^- C, -\-D, + A, = /„ (2)
and A,+B,+B.-\- C, + €, + £>, -j-D,-j- A.- 360° = /, (3)
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544 SURVEYING.
If the angles have all been equally well observed, — that is, if
their mean observed values have equal credence, — then they
are said to have equal weight, and any. residual arising from
any combination of angles should be distributed uniformly
among the angles forming such combination.* Thus /, arises
from the angles A,, B^. C,, and 2?,- This residual should there-
fore, be divided equally between these four angles. When
this is done we have
Similarly
_(ft+^)-(<:.+|)+A-^+^.-| = <. . (5)
It is evident that if /, be now divided uniformly among the
eight observed angles, it will not affect the two adjustments
already made ; neither have the adjustments already made
affected the third condition, expressed by eq. (3), since equal
amounts have been added and subtracted. Hence these ad-
justments may be made in sequence as well as simultaneously,
and we shall have for the total corrections for angle-equa-
tions
-*.-(§ -^)+^.-(§-^)+-.-(§-^)+c
-(i-:^+^.-(^^+.i)+A-(i+i)+i>. .
* The errors in the mean observed values of the angles are supposed to re-
sult from the small incidental errors and approximations made in pointing,
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6E0DETIC SUA'VEVmd.
$45
Or if V,, v^, v^, etc., be the total corrections to the several mean
observed angles for angle-equations, we have
«/. = «/.= - — ^-»
/. - 2/,
^»=^*= - — 8 — '
V, = «;, =
8 '
/, + 2/,.
8 '
(7)
368. The Side -equation Adjustment.— In the quadri-
lateral shown in the figure, let AB be the known side, and CD
the required side, which is to be computed through two inde-
pendent sets of triangles. Let ^/, ^/, 5,', etc., be the several
angles corrected for angle conditions by the corrections found
in eq. (7).
As computed through the first set of triangles, we have
_ ^Csin B^ _ AB^xwA^^xwB^
^^" sinZ?/ "" sin C sin A' . . . W
^Dsin^/ ^^sin^/sin^/
Similarly Z?C = ,;„ /-— == cin r "cTrTTv"* • • v9)
sin C/
sin C/ sin /?/
Whence
sin /4/ sin B^ _ sin -ff/ sin A^ ^
sin (7/ sin D^ "" sin C/ sin Z?/ '
reading, etc. ; in other words, they are supposed to be errors of observation and
not instrumental errors, these latter having been eliminated by the method of
making the observations. Since the sources of the errors of observation are
the same for small as for large .ingles, it follows that they should be credited
with equal portions of the aggregate error of any combination of angles, re-
gardless of the size of the angles themselves.
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54^> SURVEYING,
sin Al sin B^ sin C/ sin 2?/ __
sin 5/ shrC"sin A' sin ^/ "" ^' * * ^^^J-
or
which is called the side-equation.
It is evident that in any c^se where the angles have all
been observed, even after they have been adjusted for the
angle-conditions, this equation will not hold true, the value of
the left member being a little more or less than one. When
put into the logarithmic form for computation, therefore, we
will have
log sin -^/ + log sin B^ -\- log sin C/ + log sin /?/
—log sin B^ — log sin C/ — log sin 2?/ — log sin A^ = /^, (i i)
where /^ is the logarithmic residual due to erroneous observa-
tions.
We must now distribute this residual /^ among the log sines
according to the most probable manner of the occurrence of
the errors which caused it. For a given small error, as i', in
any angle, the effect on the log sine is measured by the loga-
rithmic tabular difference for i* for that angle. . This tabular
difference varies for different angles, being large for angles
near zero or i8o°, and small for angles near 90*^.
Let 7'/, v^\ v^\ etc., be the corrections to be made to the
angles A,\ BJ, BJ, etc., for the side-equation (i i), and let ^,.
^v ^a» ^tc-» be the corresponding logarithmic tabular differences
for i^
Now, the influences on /, of the small angular errors were
in direct proportion to the tabular differences of the correspond-
ing log sines ; therefore the corrections should be in proportion
to the corresponding tabular differences. In other words, the
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GEODETIC SURVEYING, ^ij
corrections are weighted in proportion to their tabular differ-
ences.* We therefore have the numerical relation :
V,' : d, :: < : d^ :: z^,' : ^„ etc.,
or, paying attention to signs,
But since the log-sine correction is the angular correction
multiph'ed by the tabular difference, and since the sum of these
would equal /^, we would have
^.X-«+^.'<-<^«-H'.X-^X+<^-«'.X=-A. • (13)
.From equations (12) and (13) we are to find the side-equation
corrections z^/, v^^ v^'j etc.
Dividing eq. (13) by eq. (12), term by term, we have
= _^i^ = _lM = _A^..=, . etc
* To illustrate this principle more fully, let us suppose that for a change
of i' in the angles Ax and A^ the corresponding changes in the log sines are i
for A\ and 2 for A%\ then for a given error of i in log sin ^1 + log sin Ai — I
there are two chances that it came from A\ to one chance that it came from A^
when these angles were equally well observed. If the error is to be idivided
between the angles Ax and A%, therefore, we should make the correction to A%
twice as great as the correction to Ax^ or vi': v»': z </i: </», whence -j^ =-^ •
ax at
The same reasoning would hold evidently for any number of angles, hence eq. (12).
t By eq. (ir) it will be seen that the corrections to the angles having odd sub-
scripts must be of opposite sign from those having even subscripts.
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548 SURVEYING,
Whence we have, for the values of these corrections,
vj _ _ V _ ^/ _ _ ^4' _ ^5' _ _ '^^ _ y/
d^ d^ d^ d^^ d^'^ ^e "~ ^T
_ < _ A
d, ~ 2id')'
(14)
We have now found a set of corrections, v„ z^,, v^, etc. (eq.
7), for the angle-equations, and a set of corrections, v^\ z//, v^\
etc. (eq. 14), for the side-equation ; but they were determined
independently and not simultaneously, and therefore, when
successively applied, each set of corrections will disturb the
former adjustment somewhat. Thus, if the corrections in eq.
(7) be first applied, and then those of eq. (14), using the par-
tially corrected angles in finding /^ by eq. (11), we would find
eq. (10) would be satisfied, but /„ /„ and /„ in equations (i), (2),
and (3), would now not be zero when the newly adjusted angles
were used. Another set of corrections z//, z//, z;/, etc., might
now be found by eq. (7) for Che adjusted angles A^", ^/, ^/,
etc , and so on by successive approximations, using the correc-
tions of equations (7) and (14) alternately, until both sets of
conditions were satisfied within the desired limits. It will
usually be found, however, that the adjustment for side-equa-
tion does not materially disturb that for angle-equations. If
the angles were all the same size, so that the corrections to the
log sines would have equal weight, the first adjustment would
remain undisturbed. In this case, the corrections for side-
equation would all be numerically equal, the odd and even
subscripts having opposite signs. If the observed angles range
between 30° and 60°, as they would in a fairly symmetrical
quadrilateral, then the errors of this approximation would be
quite inappreciable.
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GEODETIC SURVEYING,
549
369. Rig^orous Adjustment for Angle- and Side-equa-
tions.— Let the angle-equation adjustments be applied as given
by eq. (7). Then, using these adjusted angles, let the correc-
tions to the angles for side- equation be so expressed that they
shall not be inconsistent with the angle-equation conditions,
whatever their values. This may be done by letting
«'/= ^o + -^i»
^.' = — ^0 + ^„
vl = - ;ir, — x^.
< = -^0 - ^. ;
W = - ^0 + ^4 ;
W = - ^. - •^«- J
(15)
Then, analogous to eq. (13), we may write
^i(^. + ^1) - <(^« - ^1) - dlx, - ;r,) + dlx, + x^
+^t(^.+^3)-^e(-^o-~-^.)--^X-^«-^4)+^'»(-^o+^4)=-^4; . (16)
or
+ (rf3 + <K + K + ./.)^. + K + ^»K=-/4, . (17)
wherein /^ is given by eq. (11), and the ^/'s are the tabular
differences for one second for the several log sines as before.
If, for simplicity, we write for the coefficients of x^, ;r„ x^,
jr., and;r^, respectively, Co, C„ C„ C„ and C^, then (17) becomes
0^0 + C,^i + C>r, + C,^. + C,x, = - A. . . (18)
It now remains to find the values of x^^ x^, x,, x^, and x,y such
that their combinations which make up the angle-corrections as
given in eqs. (15) shall be the most probable.
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550 SURVEYING,
To make (i8) symmetrical with (15),* we may put it in the
following form:
+ (^;r.+ C,;r,) = -/,. , (19)
In the argument preceding the derivation of eq. (12) it was
found that the measured angles required to be corrected by a
series of quantities («'/,«'/, etc.), which quantities were found to
be to each other as the tabular differences of the angles them*
selves; and eq. (13), which is a summation of log sine correc*
tions, shows that when eq. 12 is true it is equivalent to saying
that the most probable set of angle corrections («/i', v4^ etc) is
that set which are respectively proportional to their numerical
coefficients (^„ ^„ etc.). This is, in fact, a general law of the
theory of probabilities ; and hence we say, in eq. (19), that the
most probable corrections (;r^, x^y etc.) are those which are
proportional to their several numerical coefficients, or, we may
write at once, as a condition of the greatest probability :
x^ix^ii I C^\ x^ix^ii I c„ etc. i
4 4
* This is done in order to reduce the weight of xo to that of each of the
other four x components of the v corrections in (15), as x^^ enters eight
times in those eqiiatioo3 while each of the others enters only twice.
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GEODETIC SURVEYING. 551
Therefore the condition of the highest probability gives
.... (20)
Dividing (18) by (20), term by term, we have
c.-^c\- c. - C3 ~ c:
^ + c,' + c: + c: + 67 = - - - = - ^
4 4^0 ^1
whence
4J^. X,
jr. X, X,
c,-c/
- r — r* . . (21
From equation (21) the side-equation corrections can be
computed, which will not disturb the angle-equation adjust-
ment, and which are the most probable corrections to the
several angle-values.
The second or rigid method will be found much more satis-
factory than the method by approximations. The complete
adjustment consists in applying to the mean measured
values, the corrections from angle-equations given by equation
(7), and then applying to these corrected angles the corrections
found by equation (21).
Note. — The results obtained in the above adjustments are
identical with those found by the method of least squares, and
the fundamental principle by which they are obtained is also
the same as that of least squares, viz.: that the arithmetic
♦ Note that 2{C') docs not include C;>».
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55^ Purveying.
mean of properly weighted observations is the most probable
result, and is identical with that obtained by making the sum
of the squares of the corrections a minimum. For least-square
solutions of this problem, see Clarke's " Geodesy," pp. 263-^,
and Wright's " Adjustment of Observations," pp. 303-8.
Example.
The following iS the numerical computation of the quadrilateral shown in
the figure. AB\^ the known side, and CD is to be found. The mean observed
values of the angles are given in the second column. The corrections for
angle-equations are given in the third column, and are the same for all three
methods of solution given above. The spherical excess is here applied only to
the quadrilateral as a whole, or to /«, thus distributing it equally among the
several angles. This is a common way of doing it, although if the excess is
considerable, and the several triangles very unequal in size, as is the case here,
it should be applied to the several triangles according to their size, as stated in
the foot-note, p. 514.
In columns 7 and 8, the corrections for side equation are worked out by the
two methods given to show the relative results. Thus, from eq. (14) we find
the values of vx\ v% , etc., for the first approximation. Applying these to the
first corrected values in column 4, and again taking out the values of A, A,
and /«, for angle-equation conditions, we find they are not zero, but very small.
It would probably be sufficient to work out a new set of angle-corrections by
eq. (7), and then consider the quadrilateral adjusted. In this example, the
final values thus found would then differ from the final values by the rigid
adjustment by not more than o'.2 for any angle.
If we compute CD from AB^ assuming the latter to be 25,000 feet in length,
we obtain 88.670.9 ft. in passing through the triangles ABC and BCD, while
if we pass through the triangles ABD and ADCvig obtain 88671. i ft., a dis-
crepancy of 0.2 ft., and giving a mean value of 88671.0 ft The discrepancy of
fo.2 ft. in the two results by the rigid solution results from not computing the
corrections beyond tenths of a second.
If simply a check on the final corrected values is desired, it may be obtained
by adding them, when their sum should equal 360* + spherical excess, or by
taking out the log sines and seeing if // in oq. (ii) is zero. In this case it is
not zero, but 9, resulting from not carrying out the corrections beyond tenths
of seconds, as mentioned above.
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GEODETIC SURVEYING.
553
Check.
Log. Sines
of ihe
Corrected
Angles.
HP
o» o> <> <>
0
6"
en r^ 00 M
r«« m 00 1^
d» d» d» o»
S
d>
r i
*•' i
V.
> ::-
>1
» =i.
. S +
i r '1
ii ~:
1 i
w« d -^
5 + ^
1 II «
"fl
1 ^
lis.
Final Corrected
Angles.
Rigid Solution
(Third Method).
°^s Si's
? ^ S' s
q»
8
8
V 6'
M 10 («. ^ V)
0' d d d d
+ + + + +
!l II II II II
I
1
o
U
If"*
>0 0 tA ^
*b 0 d 0
+ + + +
<«• 00 m «
0000
1 1 1 1
q
0
s
1 J
+
-5 d
II II
^ 0 fn 0
"b 0 0 0
CI » <n q
0000
1 1 1
0
1
111
« ^ M 00
•
m 00 0 >A
+ + + +
9
«U
H II n II II
% ••- "c "* %
•<
1
{/I
i
III!
6^ 6<. 0 6^
1
5
6* 6* 6* 6^
S
II II II II n
First
Corrected
Values.
00 q « M
R S' 5* S;
1^ CO q o»
q>
8
Cor-
rections
for
Angle-
equa-
tions.
"doom
1 1 1 1
t>. rn 00 c<
0 0 0 M
till
q»
1
II
"^ *? T *?
*)* m 00 ^
^ M 00 M
00
8
0* 0
0 5'
+ +
u II n
II i ff
1
<CQUS
of jqi
§
11
•0
34
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554 SMP'£y/j\rA
ADJUSTMENT OF LARGER SYSTEMS.
370. Used only in Primary Triangulation. — The simul-.
taneous adjustment of all the angles in an extended system of
triangles with one measured base which is taken as exact, is a
very complicated problem. The methods of least squares must
here be applied, so that a discussion of this problem belongs
rather to a treatise on geodesy than to one on surveying. The
adjustments of a triangle and of a quadrilateral will be found
sufficient for all secondary work, or such as is intended to serve
only for topographical or geographical purposes. Especially
is this true if the stations be so selected that the observed lines
will form a series of quadrilaterals. The adjustment of these
quadrilaterals by the rigid method given above gives nearly
as good results as could be obtained by reducing the work as a
single system. For a discussion of the least-square methods
of adjustment of an extended system of triangles the student
is referred to ** Primary Triangulation of the U. S. Lake Sur- '
vey," being Professional Papers, Corps of Engineers U. S. A.,
No. 24; Report of the U. S. Coast and Geodetic Survey for
1875, Clarke's Geodesy; and especially to Wright's "Adjust-
ment of Observations."
The facility and accuracy with which base-lines may now
be measured by means of long steel tapes will result in actually
measuring many more lines than has heretofore been done, and
so errors from angular measurements will not be allowed to
accumulate to any great extent. It is not improbable that
geodetic methods will be materially influenced by this new
method of accurate measurement.
371. Computing: the Sides of the Triangles.— After the
angles of the system are adjusted, the sides of the triangles are
computed by the ordinary sine ratio for plane triangles. If
the system consist of simple triangles, then one side is known
and the other two sides computed from it. In this case there
is no check on the computation except what the computer
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GEODETIC SURVEYING. 555
carries along with him, or what may be obtained from a dupli*
cate computation.
If the system be made up of a series of quadrilaterals, then
the line which is common to two successive quadrilaterals is
computed through two sets of triangles from the previous known
side. Thus if the quadrilateral of Fig. 142 be one of a series,
the lines in common being AB and CD, then AB is computed
in duplicate from the previous quadrilateral, and the mean of
the two results taken. In the triangle ABD compute AD, and
then in the triangle ADC compute DC\ in the triangle ABC
compute BC and then in the triangle jSCZ? compute Z^C again*
There are thus obtained two independent values of DCy as
computed from AB, If the adjustment had been exact these
values would have agreed exactly, but the adjusted angles
were computed only to the nearest second, or tenth of a second ;
hence the two values of DC will agree only to a corresponding
exactness. If the system be composed of quadrilaterals and
the adjustment be made to the nearest second, then the two
values of DC would probably differ in the fifth or sixth signifi-
cant figure. If the adjustment be made, to the nearest tenth
of a second, and a seven-place logarithmic table be used, then
the two values of DC should begin to differ in the sixth or
seventh place. Of course the adjusted values are not the true
values of the angles, but simply the most probable values. If
the angles were not accurately measured the adjusted values
may still be considerably in error, but any such errors would
not prevent the two values of CD from agreeing, since this
agreement is one of the conditions which the adjustment is
made to satisfy. The disagreement between the two computed
values of CD comes only from the inexactness of the computed
corrections to the angles, an angle, like a length, being an in-
commensurable quantity, and hence some degree of approxi-
mation is necessary in its expression. If the true computed
values of CD differ by more than the amounts above signified,
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556
SUHrEYING,
then it is probable that an error has been made in the com-
putation.
Such computations as the above are best arranged in the
following form :
Sta-
tions.
D, ..
A ...
B^..
C, ..
Z>...
A^..
C, ..
B...
Ax ..
Z>. ..
C ...
B,..
Adjusted Angles.
Log. Sines.
Log.
Disunces.
DisUnces
in Feet.
7,923.32
18,108.19
28,102.77
7.923 32
17,279.67
Sides.
iS** 49 37". 5
■
47 31 19 .3
39 48 06 .1
83 26 II .3
9.5088166
a. €.(.4911834)
9.8677843
9.8062698
a. €.(.1937302)
9.9971441
3.8989072
4.2578749
Base^^»
AD
4.4487492
CD
13 20 29 .0
30 12 52 .4
9.3631467
a. €.(.6368533)
9.7017747
3.8989072
4-2375352
hAseAB
BC
37 56 05 .7
88 55 19 .6
9.7887098
a. €.(.2112902)
9.9999229
4.4487483
28,102.71
CD
Here the angles are written in such an order that when the
arithmetical complement (a. c.) of the subtractive log. sine is
taken out, the three logs, will be in convenient relative positions
for adding. This will become evident on a study of the table.
In computing the second and fourth triangles it is evidently un-
necessary to write the log. distances again in their proper lines,
since they already stand conveniently just above. It will be
noted that in the above form the diagonals of the quadrilateral
were not computed.
When a series of triangles in a chain are to be computed
from a base-line and the adjusted angles, whether these form a
single row, as in I., p. 474, or a double row, making complete
quadrilaterals, as in III. of the same page, it is customary to
compute all the sides ; and then, in case of a series of quadrilat-
erals, the average length (or log.) of the sides common to two
♦ The length of ihc (jiasgow Hase, p. 522, is here used.
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GEODETIC SURVEYING,
557
adjacent quadrilaterals is used as a new base for the next one.
Thus, if all the sides in the above quadrilateral were to be
computed, and if the side CD be computed through the two
diagonals instead of through the outer sides, the computations
would be arranged as follows :
Sta-
tions.
C ...
B,..
A...
D...
A ...
B...
C ...
D...
B...
D...
C...
A ...
Adjusted Angles.
Log. Sines.
Disunces.
Distances
in Feet.
Sides.
13* 20' 29".o
136 26 38 .9
30 12 52 .4
18 49 37 .5
"3 39 03 .7
47 31 19 -3
53 08 35 .1
37 56 05 .7
88 55 19 .6
9.3631467
a. €.(.6368533)
9.8382579
9.7017747
3.8989072
4.3740184
4.2375352
3.8989072
4.3519888
4.2578749
7,923 32
23,660.20
17,279.67
7,923.32
22,449.09
18,108.19
Base AB
AC
BC
Base AB
BD
AD
9.5088166
a. €.(.491 1834)
9.9618982
9.8677843
9.9031638
a. €.(.0968362)
9.7887098
9.9999229
4.3519888
4.2375348
4.4487479
22,449.09
17,279.65
28,102.69
BD
BC
CD
56 45 43 .2
39 48 06 .1
83 26 II .3
9.9224146
a. €.(.0775854)
9.8062698
9.9971441
4.3740184
4.2578736
4.4487479
23,660.20
18,108.14
28,102.69
AC
AD
CD
In this form the logs, of the angles and of the sides opposite
to them are placed on the same line, the known side and its
angle opposite being always written on the first line, and the
a. c. of this log. sine taken out a#shown. The log. distances of
another side of any triangle is then the sum of its corresponding
(opposite) log. sine, the a. c, and the log. distance of the known
side. These sums are readily taken without copying off the
figures, and they are written at once in the log. distance column.
The original observed, but uncorrected, mean angles are com-
monly given in addition to the corrected angles, but these have
been omitted here as not essential to explain the form. In the
above computation it so happens that the log. side CD is the
same in each case. Had they differed, as in the previous tabu-
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558 SURVEYING.
lation, the mean log. would be used as the log. of this known
side for the next set of triangles.
i-^ -^^ ^ ^ LATITUDE AND AZIMUTH.
372. Conditions. — In the methods here given for obtaining
latitude, azimuth, and time, the instrument used may either be
an ordinary field transit mounted on its tripod, or a more elabo-
rate altazimuth instruftient, such as shown in Figs. 132 and 134.
The accuracy sought is only such as is sufficient for topographi-
cal or geographical purposes. Both the field methods and the
office reductions are of the simplest character; but all large
errors are eliminated, so that the results will be found as accu-
rate as it is possible to obtain with anything less than the regu-
lar field astronomical instruments. This higher grade of work
falls within the sphere of the astronomer rather than of the
surveyor.
373* Latitude and Azimuth by Observations on Cir-
cumpolar Stars at Culmination and Elongation. — When
latitude and azimuth are to be found to a small fraction of a
minute, or as accurately as can be read on the instrument used,
if this be an ordinary field transit, the most convenient method
is by means of observations on circumpolar stars. The observa-
tion for latitude is made on such a star when it is at its upper
or lower culmination, since it is then not changing its altitude,
and the observation for azimifth is made at elongation, since
then the star is not changing its azimuth. At these times a
number of readings may be taken on the star, thus eliminating
instrumental constants by reversals, since a half hour may be
utilized for this work without the star sensibly changing its
position so far as the use it is serving is concerned. Two close
circumpolar stars have been chosen whose right ascensions
differ by about five hours and thirty minutes. They therefore
always give a culmination and an elongation about thirty min-
utes apart. This is very convenient, since it allows observations
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GEODETIC SURVEYING.
559
to be made for latitude and azimuth at one setting with a suf-
ficient intervening interval to complete one set of observations
before commencing the next.
The two stars selected are Polaris {a Ursae Minoris), which is
of the second magnitude, and 51 Cephei, which is of the fifth
magnitude. Their relative positions are shown in Fig. 143.
MiNVpouma)
Fig. 143.
The position of 51 Cephei may be described with reference
to the line joining "the pointers," in the constellation of the
Great Bear, with Polaris. Thus, 5 1 Cephei is to the right of
this line, when looking towards the pole-star along the line, at
a distance of about three times the sun*s disk from the line, and
of about y?z/^ times the sun's disk from Polaris, in the direction
of the pointers.
In case 5 1 Cephei is not visible to the naked eye, as it may
not be on moonlight nights, or with a slightly hazy atmos-
phere, it may be found, when near elongation, by the tele-
scope, as follows :
Having carefully levelled the instrument, turn upon Polaris.
When 51 Cephei is near its eastern elongation Polaris is near
its upper culmination, and when near its western elongation
Polaris is near its lower culmination. To find 51 Cephei at
eastern elongation, therefore, after taking a pointing on Pola-
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560
SURVEYING.
ris, lower the telescope (diminish the vertical angle) by about
one degree (if the time is about twenty minutes before elonga-
tion), and then turn off towards the east about two and a half
degrees. This will bring the cross wires approximately upon
the star.
To find it at western elongation, simply reverse these angles ;
that is, increase the vertical angle one degree, and turn off to
the west two and one half degrees.
The following table gives the times of the elongations and
culminations of these two stars for 1901 for latitude 40°, which
may be used for observing azimuth and latitude. The times
given are for the nights following the dates named in the first
column.
TIMES OF ELONGATION AND CULMINATION. 1901.*
LATITUDE, 40*.
Date.
Polaris (o
Urs. Min.).
51 Cbphei.
Elon-
tion.
Time.
Cul-
mina-
tion.
Time.
Elon-
tioo.
Time.
Cul-
mina-
tion.
Time.
Jan. 1
W
ia*»34".8 A.M.
U
6*39-.8 P.M.
W
5b59-
.9 A.M.
U
13^14". I A.M.
Feb. 1
"
10 33 .4 P.M.
L
4 35 .5 A.M.
»«
3 57
•9 "
*•
10 8 .3 P.M.
Mar. 1
»'
8 41 .9 *'
"
a 45 .0 **
»♦
a 7
.8 "
»«
8 18 .1 **
April 1
"
♦639 .8 "
♦*
la 4a .9 '*
ti
la 9
.6 "
•'
•6 16 .0 "
May 1
E
*4 48 .IA.M.
•♦
10 43^.1 P.M.
*♦
10 7
.5 P.M.
L
4 IS .8 a.m.
June 1
•♦
2 46 .7 **
"
8 43 -6 *•
•*
•8 5
.5 "
It
s 14 .0 '•
July 1
**
la 49 .a "
'*
♦6 46 .1 "
E
•6a4
.a A.M.
"
13 15 .8 "
Aug. 1
"
10 47 .8 P.M.
U
4 42 .7 A.M.
**
4 aa
•4 "
*»
10 14 .t P.M.
Sept. 1
Oct. 1
..
8 46 .4 "
•648 .7 "
u
a 41 .3 •*
la 43 .6 *•
..
a 20
13 33
.8 •♦
.1 "
"
8 13 .3 *•
♦6 14 .7 "
Nov. I
w
4 36 .7 A.M.
•♦
10 41 .9 P.M.
"
10 at
.5 i-.M.
U
4 II .1 '*
Dec. I
**
a 38 .6 "
**
8 43 .6 "
8 33
.7 '*
a 13 .4 "
* Probably not visible to the naked eye.
From the above table it is evident that both an elongation
and a culmination of one of these stars can always be obtained.
For other days than those given in the table, either inter-
* For a table of computed azimuths of Polaris when at elongation from 1S95
to 19TO and for latitudes between 25* and 50*, sec page 33, , .
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GEODETIC SURVEYING.
561
polate, or find by allowing 3'".94 for one day, remembering
that each succeeding day the elongation occurs earlier by this
amount.
For other years than 1901, take from the table the time cor-
responding to the given month and day, and add for Polaris
o"'.4, and for 51 Cephei o".5 for each year after 1901; also,
Add I" if the year is the second after leap-year.
Add 2*" if the year is the third after leap-year.
Add 3™ if the year is leap-year before March I.
Subtract i"* if the year is leap-year after March i.
For the first year after leap-year there is no correction ex-
cept the periodic ones of o'".4 and o".5 per annum.
For other latitudes between 30° and 50° north latitude cor-
rect the times of elongation as follows:
For each degree south of 40°,
Add to the western or subtract ) _, ,.__ \ o™.l4 for Polaris.
( o".29 1
from the eastern time of
For each degree north of 40°,
Subtract from the western or add
to the eastern time of
for 51 Cephei.
> elongation
) . jo". 18 for Polaris.
\ «l<>"g^*»o^ io«".39 for 51 Cephei
The following table gives the pole distances of Polaris for
Jan. I of each third year from 1900 to 1930:
POLE
DISTANCE
(90--
DECLINATION) OF POLARIS
1900.
1903.
i«»ia'.6
1906.
i*ii'.7
1909.
1912.
1915.
1918.
1931.
1994.
1927.
1930.
i*»io'.8
1V.8
i«»8'.9
i«»8'.o
i»7'.o
i«»6'.i
iV.»
1V.3
Interpolate in the above table for the first of January of
intermediate years. For other months than January of every
year, add to the pole-distances found for January the following
corrections: Feb. o'.i; Mar. o'.2; Apr. 0^.3; May o'.5 ; June
o'.6; July o'./; Aug. o'.6; Sept. o'.5 ; Oct. o'.3; Nov. o'.2;
Dec. o'.i.
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562
SURVEYING.
To observe for latitude no knowledge of the geographical
position is needed.
374. The Observation for Latitude consists simply in
observing the altitude of a circumpolar star at upper or lower
culmination and correcting this altitude for the pole distance
of the star and for refraction.
Let
0 = latitude ;
d = polar distance ;
r = refraction ;
// = altitude ;
then
0 = A q: d—r;
(0
the minus sign being used for upper, and the plus sign for
lower, culmination observations. The value of r is taken from
the following table of mean refractions computed for barometer
30 inches, and temperature 50° F.
TABLE OF MEAN REFRACTIONS.
Altitude.
Refraction.
Altitude.
Refraction.
10"
5' 19"
1 20^
2' 39"
II
4 51
25
2 04
12
4 28
30
I 41
13
4 07
35
I 23
14
3 50
40
I 09
15
3 34
45
0 58
16
3 20
50
0 ^
17
3 08
1 60
0 34
18
2 58
70
0 21
19
2 48
80
0 10
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GEODETIC SURVEYING, 563
The index error of the vertical circle is eliminated by read-
ing with the telescope direct and reversed, providing the verti-
cal circle is complete. If the vertical limb is but an arc of 180°
or less, the index error cannot be eliminated in this way. In
this case the second method is recommended.
375. First Method. — Mount the instrument firmly, pre-
ferably on a post, and adjust carefully the plate-bubbles,
especially the one parallel to the plane of the vertical circle.
About five or ten minutes before the star comes to its culmi-
nation read the altitude of the star with telescope direct.
Revolve the telescope on its horizontal axis and also on its
vertical axis, relevel the instrmnent if the bubbles are not in the
middle, but do not readjust the bubbles, and bring the tele-
scope upon the star. Make two readings in this position.
Revolve the telescope and instrument again about their axes,
relevel, and read again in first position. This gives two direct
and two reversed readings taken in such a way as to eliminate
the error from collimation, the index error of vertical circle,
and also the error of adjustment of the plate-bubbles. The
result, when corrected for refraction and the pole distance of
the star, should be the latitude of the place within the limits
of accuracy and exactness of the vertical circle-readings.
376. Second Method. — An "artificial horizon,'* formed by
the free surface of mercury in an open vessel, may be used in
conjunction either with the transit or a sextant. If the former
is used two pointings are made — one to the star and the other
to its image in the mercury surface. The angle measured is
then twice the apparent altitude of the star. The position of
the vessel of mercury will be on a line as much below the
horizontal as the star is above it. The instrument is first set
up and then the artificial horizon put in place. The surface
of the mercury must be free from dust. If the mercury is not
clean it may be strained through a chamois-skin or skimmed
by a piece of cardboard. Any open vessel three or more
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5^4 PURVEYING.
inches in diameter may be used for holding the mercury. It
should be placed on a solid support and protected from the
wind.
The observations with a transit would then consist in taking
a reading on the star just before culmination, two readings on
the image, and then one on the star. The index error of the
vernier on the vertical circle will then be eliminated, since both
plus and minus angles have been read, and their sum taken for
twice the altitude of the star. This method is adapted to
transits with incomplete vertical limbs.
The Sextant may also be used with the artificial horizon
and will give more accurate results than can be obtained with
the ordinary field transits. The double altitude angle is then
measured at once by bringing the direct and reflected images
of the star into coincidence. In both cases the observed angle
is 2A, and the latitude is found from equation (i), as before.
If there is much wind the mercury basin may be partially
covered, leaving only a narrow slit in the vertical plane through
instrument and star, or the regular covered mercurial horizon
may be used. This is covered by two pieces of plate-glass set
at right angles to each other like the roof of a house. If the
opposite faces of these glasses are not parallel planes, an error
is introduced. This is eliminated by reversing the horizon
apparatus on half the observations. It is best, however, to
avoid the use of glass covers, if possible.
If tin-foil be added to the mercury an amalgam is formed,
whose surface remains a perfect mirror, which is not readily
disturbed by wind. As much tin-foil should be used as the
mercury will unite with. Observations may then be made in
windy weather without the aid of a glass cover.
377. Correction for Observations not on the Meridian.
— If the star is more than five or ten minutes of time from the
meridian, it is necessary to apply a correction to the observed
altitude to give the altitude at culmination. The following
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GEODETIC SURVEYING, 565
approximate rule gives these corrections for the two circum-
polar stars here used, with an error of less than \" of arc when
the observation is taken not more than 18 minutes of time
from the star's meridian passage, and the error is less than 10"
oiarc when the observation is made 32 minutes of time from
the meridian.
Rule for reducing circummeridian altitudes to the altitude at
culmination.
For Polaris: Multiply the square of the time from meridian
passage, in minutes, by 0.0444, and the product is the correc-
tion in seconds of arc.
For 51 Cephei: Multiply the square of time from meridian
passage, in minutes, by 0.1017, and the product is the correc-
tion in seconds of arc.
The correction is to be added to the observed altitude for
upper culmination, and subtracted for lower culmination.
By using these corrections an observation for latitude may
be made at any time for a period of about one hour, near the
time of culmination.
378. The Observation for Azimuth is made on one of
the two stars here chosen when it is at or near its eastern or
western elongation, for the same reason that latitude observa-
tions are taken at culmination. The azimuth of a star at
elongation is found from the formula.
. . , sine of polar distance , ,
sine of azimuth = : 7-. ^. , — . . . (i)
cosine of latitude ^ '
This formula is so simple that it is hardly necessary to give a
table of values of azimuths for various latitudes. Such a table
is given for Polaris, however, on p. 33. The pole distances
are given on p. 561, and the latitude is found by observation.
It is not necessary to know the azimuth of the star at elonga-
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566
SURVEYING.
tion before making the observations. This can be computed
afterwards from the observed latitude.
The observation for azimuth consists simply in measuring
the horizontal angle between the star and some conveniently
located station, marked by an artificial light.' The operation
is in no sense different from the measurement of the horizontal
angle between two stations at different elevations. The great
source of error is in the horizontal axis of the telescope. If
this is not truly horizontal then the line of sight does not de-
scribe a vertical plane, and since the two objects observed have
very different elevations, the angle measured will not be that
subtended by vertical planes passing through the objects and
the axis of the instrument. To eliminate this error the tele-
scope is reversed, and readings taken in both positions. The
following programme is recommended:
PROGRAMME FOR OBSERVING FOR AZIMUTH ON A CIRCUM-
POLAR STAR AT ELONGATION.
Insirument.
Time of Observation.
Reading on
Direct
10 min. before elongation.
Mark.
Siar.
it
Mark.
Reversed
<<
Direct
7 «* *•
4 •* "
2 ** "
2 min. after **
it
4 •• "
7 " •'
lo ** **
«(
Reversed. ... ...
The instrument should not be relevelled nor the bubbles
adjusted after the observations have begun. If the instrument
should be disturbed of course the series is spoiled. If the change
of level is gradual, it and all other errors will be eliminated except
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GEODETIC SURVEYING, 567
those of graduation. Of course both verniers are to be read
each time.*
Having found the latitude, the azimuth of the star at elon-
gation is found from equation (i) above. This is then added
to or subtracted from the horizontal angle between mark and
star, as the case may be, to give the azimuth of the mark from
the north point. If the azimuth is to be referred to the south
point, which it generally is, we must add or subtract 180°.
379. Corrections for Observations near Elongation. —
As in the case of observations for latitude, we may have an
approximate rule for reducing an observed azimuth when near
elongation to what it would have been if taken at elongation.
The limits of accuracy are also about the same, but the factors
are slightly different.
Rule for reducing azimuth observations on Polaris and 5 1
Cephei near elongation to their true values at elongation, for
latitude 40°.
For Polaris, multiply the square of the time from elonga-
tion in minutes by 0.058, and the product will be the correction
in seconds of arc.
For 51 Cephei, multiply the square of the time from elonga-
tion in minutes by 0.124, and the product will be the correction
in seconds of arc.
The formula for reduction, when near elongation, is
er= 112.5 /' sin i" tan A^
where c = correction to observed azimuth in seconds of arc ;
/ = time from elongation in seconds of time;
A = azimuth of star at elongation.
log 112.5 sin 1" = 6.7367274.
From this formula and that of equation (i) we may compute
the coefficients for the above approximate rules for any latitude.
♦ If a mercury surface be used and alternate readings be taken on the star and
on the image, all errors from inclined horizontal axis are eliminated, and extremely
accurate work can be done with an ordinary transit.
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568 SURVEYING.
Thus, for latitude 30° we have azimuth of Polaris, 1885, 1°
3o'.4, whence the coefficient of reduction for elongation of
Polaris in latitude 30° is found to be 0.052, and for latitude 50®
it is 0.069.
For 51 Cephei, this coefficient for latitude 30"^ is o.lio, and
for latitude 50°, 0.148.
From the above data the corrections for an observation of
a circumpolar star near elongation may be computed.
If azimuth be reckoned from the south point, as is common
in topographical and other geodetic work, and if it increase in
the direction S.W.N. E., then a star at western elongation has
an azimuth of less than 180°, and at eastern elongation its
azimuth is more than 180°.
The corrections to reduce to elongation, as above com-
puted, should be added to the computed azimuth of the star at
western elongation, and subtracted when at eastern elongation.
380. The Target. — This may be a sort of box, in which a
light may be placed. A narrow vertical slit should be cut, sub-
tending an angle, at the instrument, from one to two seconds of
arc. This should be set as far from the instrument as conven-
ient, as from a quarter of a mile to one mile. The width of
slit desired may be computed for any given angular width
and distance by remembering that the arc of one second is
three-tenths of an inch for a mile radius. The target should
be sufficiently distant to enable it to be seen with the stellar
J focus without appreciable parallax, as the instrument should
• not be refocused on the target. This target may be set on
any convenient azimuth from the observation-station, as upon
one triangulation station when the observations are taken at
another, thus obtaining directly the azimuth of this line.
381. Illumination of Cross-wires.— Various methods are
used to illuminate the wires, the crudest of which is, perhaps, to
hold a bull's-eye lantern so as to throw light down the tele-
scope-tube through the objective, taking care not to obstruct
the line of sight.
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GEODETIC SURVEYING. 569
A very good reflector may be made from a piece of new
tin, cut and bent as in Fig. 144. The straight
strip is bent about the object end of the tele-
scope tube, leaving the annular elliptic piece
projecting over in front. This is then bent to
any desired angle, preferably about forty-five
degrees, and turned so that an attendant can
reflect light down the tube by illuminating the disk from
a convenient position. This position should be so chosen
that the lantern may throw the light from the observer,
rather than towards him. If the reflecting side of the disk be
whitened, the effect is very good. The opening should be about
three-fourths or seven-eighths inch in its shorter diameter, the
longer diameter being such as to make its vertical projection
equal to the shorter one. There is, of course, no necessity of
limiting or of making true the outer edges of the disk.
^ 381a. Azimuth and Latitude from Polaris at any Hour.*
"' — This method consists in the use of a few simple tables by
which the azimuth of Polaris and its altitude above or below
the pole are found in terms of the hour-angle of the star.
The horizontal circle can be clamped at the computed azimuth
for any chosen time of observation, and when the star is on the
cross-wires at this instant, the horizontal circle is oriented,
and the vertical-circle reading has but to be corrected by a
single addition or subtraction to give the true latitude. It is
first necessary to find the hour-angle of the star.
The epochs when the star and the mean sun are on the
meridian together are given in Table I., p. 569^, for each year
from 1900 to 1930. Assuming that the observer's watch is set
for local mean solar (or clock) time, instead of standard (hourly
meridian) time, then the watch shows on its face, any time be-
fore midnight, the hour-angle of the mean sun. (From mid-
night to noon the hour-angle is 12 more than the clock-hour.)
But the star comes to the meridian 4 min. earlier than the sun
* Method used by Prof. G. C. Comstock, Univ, of Wis.
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569a
SURVEYING,
Table I. — Epochs.
(When sun and polaTis come to the oieridian together.)
Year.
Epoch.
Year.
Epoch.
Year.
Epoch.
Year.
Epoch.
1901
igoa
•903
»904
T906
X907
»9o8
April.
12.9
12.6
12.9
13.6
13.0
1909 ... r --
April.
13^8
»9>7
1918
1919
1920 . ...
1921
1922
1923
»924
April.
14. 1
14.2
:m
15.1
'4.5
«9«5
1926
J9a7
»928
«9»9
«93o
»93i
1912
April.
1910 ..
1911 ..
1912 ..
IQI^. . . .
1914 .
IQie. ...
1916...
Table II.— Z', for Year and
Latitude.
Lat.
20«
K
4o«
SO**
1900
0.8a
o.8i
1.00
1. 10
19Z0
0 78
0.96
1.14
1920
»930
0.75
0.81
0.92
1.09
0.72
0.77
0.87
1.04
Table IV. — Hour-angle Correc-
tions.
/ = local mean time + 4«(date - epochXx - ,^^).
Az. = i8o*» + F^a,
Table III.— ^a for Year.
Year.
•
1900
x.oo
1910
1920
1930
0 96
0.92
0.87
/
a = az. cor.
b = lat. cor.
/
Hours.
Hours.
0
- 0' -
-
-74'-
«4
z
-
- 7a -
as
a
-
-64 -
aa
3
-*»Tc-
-
- 5a -
ai
4
-
- 37 -
ao
1
:
- 19 -
— 0 4-
:i
7
-J^li-
-
+ 19 --
17
8
9
— 82 ° -
-
\xtt
16
15
10
11
:lsl
_
^%%"
Z4
IX.
L_
1 +72 --
za
Lat. = Altiiudc -f F^b,
The signs on the left go with the lefuband
arjfumeni, and those on the right with the right-
hand argument. Refraction Is inciuded
each day (really 3.94 min.), hence the hour-angle of the star at
any time is that of the sun plus, in minutes, four times the
number of intervening days, after the epochs given in Table I.
(More nearly it is one seventieth of this product less.) In
these tables the day begins at noon of the given date. Thus
in Table I., the epoch April 12.2 is April 12, 448 P.M., while
April 12.9 is really April 13, 9:36 A.M. If an observation is to
be made at 9 p.m. of June 11, 1901, the number of intervening
days after April 12.5 is (to the nearest tenth) 59.;^; This num-
ber times 4 min. (less ^ of itself) is 236 minites = 3 hr.-56
min. This added to the local time gives 12 hr. 56 min.
For Azimuth enter Table IV. with this argument and find
a = + 23'. But Az. = 180° + F,a, If the latitude be 43",
then from Table II. we have for 43° and 1901, F, = 1.05, hence
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GEODETIC SURVEYING. 569^
F^a = 24', and -^ = 180° 24'. That is, Polaris was then 24'
east of north. If the horizontal circle be clamped with vernier
A reading 180° 24' (the circle being graduated in the direction
S., W., N., E.), then when the star is on the cross-wire s at
9 P.M., local time June 11, 1901, in latitude 43^ the instrument
is oriented, or a zero reading gives a south pointing.
The latitude may be found by adding FJb to the altitude, as
read on the vertical circle. This would be for this case 1° 10'
greater than the altitude of the star.
It will be noticed that the time chosen was most unfavor-
able for the azimuth observation, since this was then changing
24' to the hour, while it was most favorable for the latitude
observation, as this was only changing 2' to the hour. The
most favorable time for each result is evidently when its cor-
rection, in Table IV., is varying most slowly. Even when vary-
ing most rapidly these results are obtained by this method with
an accuracy of about one minute of arc.
381b. Azimuth from Polaris at any Hour. — In the
" Manual of Instructions," issued by the Commissioner of the
General Land Office in 1901, there appeared a new set of tables
designed to enable observations for azimuth to the nearest
minute to be made at any hour by an observation on Polaris.
These tables are condensed into Table XII.
By the use of this table an observation for azimuth, of suf-
ficient accuracy for ordinary purposes, can be made at any time
when Polaris is visible.
Considering the two pages as composing one table, the two
middle columns give the time of upper culmination of Polaris
for any day of the year 1901. For other years add o°*.3 for
each year after 1901 ; also, add 1" if the year is the second
after leap-year ; add 2" if the year is the third ^tfter leap-year ;
add 3" if the year is leap-year before March i ; subtract i" if
the year is leap-year after March i.
For the first year after leap-year there is no correction ex-
cept the periodic one of 0.3™ per annum.
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570 ^VRVEYWG,
The table is arranged for giving the azimuth of Polaris at
any point in its path, the argument being the time which has
elapsed since its last upper culmination. The upper culmina-
tion sought is therefore the last one preceding the time chosen
for the observation.
Suppose this time to be August 9, 1903, at 9 P.M. By refer-
ring to the two central columns of the table, we see at a glance
that in August Polaris culminates some fifteen or sixteen hours
after mean noon, or at about 3 or 4 o'clock the next morning.
The culmination sought, therefore, is the one following mean
noon on the 8th. We therefore wish to find the time of cul-
mination of Polaris after mean noon on August 8, 1895.
We have, from the table, the time of the star's upper cul-
mination :
For August I, 1901 . . . • 16** 42™./
Tab. dif. for 7 days — 27 .5
For August 8, 1901 \& I5°'.2
Correction to 1903 + 2 .6
Time up. culm. August 8, 1903 \& I7"'.8
after noon, which is 4 o'clock and 17"". 8 A.M. of August 9.
The time chosen for the observation is 9 P.M., or 16** 42".2
after the star's last upper culmination. This is called the
hour angle of the star. Evidently it has passed its lower
culmination, and is now moving upward on the eastern half of
its orbit. Since its position in its orbit with reference to the
meridian is the significant thing, we can find this by subtract-
ing 16^ 42°\2 from its period of revolution, which is a sidereal
day, or 23'' 56™. Making this subtraction, we find the star's
position to be f" I3'".8 from its upper culmination, on the east
side. Entering the table with the argument f" I3"*.8 for the
year 1903, and for latitude 40°, we find we must interpolate
between 1° 28' and 1° 3^^ which gives us the azimuth i** 29',
which is the true azimuth of Polaris at the time of observation.
Furthermore, the table shows us that to change the azi-
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GEODETIC SURyEvrnG, 57 1
muth at this time by i' requires a lapse of time of from seven
to eight minutes.
When near the culminating points the star moves much
faster in azimuth, and it here requires but 2™"- to produce a
change of azimuth of i'. It is evident, therefore, that if the
local time is known to within one or two minutes, the method
will always give the azimuth to the nearest minute of arc.
It must be remembered, also, that it is the local time which
must be used as the time of the observation, and not the
" standard " time, now universally used in America.
By the peculiar and ingenious arrangement of this table,*
all the data necessary to make an observation for azimuth at
any hour of any day or year, until 19 12, are presented on two
opposite pages. Never before has this matter been so sim-
plified. It is usually very inconvenient to await the time of
elongation of Polaris, and at times both the elongations occur
in the daylight hours. By means of this table, and where
an accuracy of one minute of arc is sufficient, the observation
can be taken at pleasure, simply noting the time, and the azi-
muth of the star may be taken out later for that particular time.
TIME AND LONGITUDE.
382. Fundamental Relations.— In all astronomical compu-
tations the observer is supposed to be situated at the centre
of the celestial sphere and the stars appear projected upon its
surface. Their positions with respect to the observer may be
fixed by two angular codrdinates. The most common plane of
reference for these coordinates is that of the celestial equator,
and the coordinates referring to it are known as Right Ascen-
sion and Declination — corresponding to Longitude and Lati-
tude on the earth's surface.
Right ascension is counted on the equator from west tow-
■* Prepared originally by Mr. J. B. Shion, of the United States General Land
Office, Washington, D. C.
35
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57^ SURVEYING,
ards east. As a zero of right ascension the vernal equinox
is taken.
Declination is counted on a great circle perpendicular to
the equator, and is called positive when the star is north and
negative when south.
In Fig. 145
P is the pole ;
Z is the zenith of the observer ;
S is the star ;
Then R. A. star = VPS = arc VE ;
Dec. star = 5^6".
These coordinates are fixed, varying only by slow changes
due to the shifting of the reference-plane.
Another system of coordinates is often used in fixing the
place of a star, namely: Hour-angle and Declination. Hour-
angle is the angle at the pole between the meridian and the
great circle passing through the star and the pole perpendicu-
Fig. 145.
lar to the equator. Hour angle will of course be constantly
changing each instant. In Fig. 145 hour-angle = ZPS.
383. Time. — The motion of the earth on its axis is perfect-
ly uniform. We obtain, therefore, a uniform measure of time
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GEODETIC SURVEYING, 573
by employing the successive transits of a point in the equator
across the meridian of any place. The point naturally chosen
is the vernal equinox.
A Sidereal Day is the interval of time between two succes-
stve upper transits of the vernal equinox over the same merid-
ian.
The Sidereal Time at any instant is the hour-angle of the
vernal equinox at that instant reckoned from the meridian
westward fromo^ to 24''. Thus, when the vernal equinox is on
the meridian, the hour-angle is o*' oT o^ and the sidereal time
is o** o™ o\ When the vernal equinox is i** west of the merid-
ian the sidereal time is i** o™0".
We have in Fig. 145
Hour-angle of ver. eq. = ZPV = 0 =: sidereal time;
Right asc. of star = VPS = a ;
Hour-angle of star = ZPS = If;
Whence 0 ^ a = H. (i)
From this equation, knowing the sidereal time and the
R. A. of the star, the hour-angle may always be computed.
When H=Oy i.e., when the star is on the meridian, 0=z a^ or,
in other words, the R. A. of any star is equal to the true local
sidereal time when the star is on the meridian. By noting the
exact time of transit of any star whose R. A. is known, the
local sidereal time will be at once known.
An Apparent Solar Dayxs the interval of time between two
successive upper transits of the true sun across the same
meridian.
Apparent or True Solar Time is the hour-angle of the true
sun.
Owing to the annual revolution of the earth, the sun's
right ascension is constantly increasing. It follows, therefore,
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574 SURVEYING,
that a solar day is lon^jer than a sidereal day. In one year
the sun moves through 24**" of right ascension. There will
be, therefore, in one tropical year (which is the interval be-
tween two successive passages of the sun through the vernal
equinox) exactly one more sidereal day than solar days ; or, in
other words, in a tropical year the vernal equinox will cross
the meridian of any given place once more than the sun will.
The solar days will, however, be unequal for two reasons :
1st. The sun in its apparent motion round the earth does
not move in the equator, but in the ecliptic.
2d. Its motion in the ecliptic is not uniform.
On account of these inequalities the true solar day cannot
be used as a convenient measure of time. But a mean solar
day has been introduced, which is the mean of all the true
solar days of the year and which is a uniform measure of
time.
Suppose a fictitious sun to start out from perigee with the
true sun, to move uniformly in the ecliptic, returning to peri-
gee at the same moment as the true sun. Now, suppose a
second fictitious sun moving in the equator in such a way as
to make the circuit of the equator in the same time that the
first fictitious sun makes the circuit of the ecliptic, the two fic-
titious suns starting together from the vernal equinox and re-
turning to it at the same moment. The second fictitious sun
will move uniformly in the equator and will be therefore a
uniform measure of time. This second fictitious sun is known
as the Mean Sun.
A Mean Solar Day is therefore the interval between the
upper transits of the mean sun over the meridian of anyplace.
Mean Solar Time at any meridian is the hour-angle of the
mean sun at that meridian counted from the meridian west
from o^ to 24''".
The Equation of Time is the quantity to be added to or
subtracted from apparent solar time to obtain mean time.
The equation of time is given in the American Ephemeris
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GEODETIC SURVEYING. 575
for Washington mean and apparent noon of each day. If the
value is required for any other time it can be interpolated be-
tween the values there given.
384. To convert a Sidereal into a Mean-time Interval,
and vice versa. — According to Bessel, the tropical year con-
tains 365.24222 mean solar days, and since the number of side-
real days will be greater by one than the number of mean solar
days, we have
365.24222 mean sol. days = 366.24222 sid. days ;
I mean sol. day = i. 00273791 sid. days.
Let /„ = mean solar interval ;
/, = sidereal interval;
it = 1. 0027379 1.
Thus
/, = /m^ = /m+ Inik - l) =/« + 0.0027379/^ ;
Im - -^ = /. - /, (l - y = /, - 0.0027304/,.
By the use of these formulae the process of converting a
sidereal interval into a mean-time interval, and vice versa^ is
made very easy. It is rendered more easy by the use of
Tables II. and III. of the Appendix to the American Ephem-
eris and Nautical Almanac, where the quantity IJ^k—\) is
given with the argument /^, and /,( i — -r 1 with the argument /.
Example. — Given the sidereal interval /, = 15** 40"" 50'.SO, find
the corresponding mean-time interval.
/, = 15^40™ 50».50
Table II. gives for 15** 40°" 2 33.996
" " " 50^.50 0.138
/m=i5 38 16.37
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576 SURVEYING.
385. To change Mean Time into Sidereal. — Referring
to Fig. 145, suppose 5 to represent the mean sun.
Then ZPS = hour-angle of mean sun = mean-time = T\
VPE = R. A. of mean sun = a, ;
6 = sidereal time.
From equation (i), p. 573,
The right ascension of the mean sun is given in the Ameri-
can Ephemeris both for Greenwich and Washington mean
noon of each date. It is called ordinarily the sidereal time of
mean nooUy which is of course the right ascension of the mean
sun at noon, since at mean noon the mean sun is on the
meridian and its right ascension is equal to the sidereal time.
Since the sun's right ascension increases 360° or 24**" in one
year, it will change at the rate of 3™ S6*.555 in one day, or
9*.8565 in one hour.
Suppose BJ = sid. time of mean noon at Greenwich;
e^ = " " " " " " the place for which
T'is known ;
L = longitude west of Greenwich.
Then e^ = e: + 9».8565Z,
where L is expressed in hours and decimals of an hour.
In this way the sidereal time of mean noon may be obtained
for the meridian of observation.
Substituting for a, its equivalent, and reducing the mean-
time interval to sidereal,
Example. -Longitude of St. Louis, & o" 49". 16 = 6*'.oi36.
Mean time, 1886, June 10, lo*" 25"* 25'.5. Required correspond-
ing sidereal time.
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GEODETIC SURVEYING, S77
From Amer. Ephem., p. 93 :
6^0' (for Greenwich) = 5** 15™ 3". 30
6.0136X9-8565 = o 59.27
e. = 5 16 2.57
T = 10 25 25.50
TXk— \), Table III., = i 42.74
6 = 15 43 10.81
It should be remarked that the quantity 59'.27 will be a
constant correction, to be added to the sid. time of mean noon
at Greenwich to obtain the sid. time of mean noon at St.
Louis.
386. To change from Sidereal to Mean Time. — This
process is simply the reverse of that for changing from mean
to sidereal time. Using the same notation as before, we shall
have
T=e-d,-{e-e,){x-'^.
Subtracting from the given sidereal time {ff) the sidereal
time of mean noon {6^, we have the sidereal interval elapsed
since mean noon, and this needs simply to be changed into a
mean-time interval.
Example. — Given 1886, June 10, 15** 43*" io*.8i sidereal
time, to find the corresponding mean time.
^ = 15 43 10.81
(as before) 6^=: 5 16 2.57
6 -^e^— 10 27 8.24
«?- B:) (i ~ 1) (Table II.) = i 4274
r= 10 25 25.50
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578 SURVEYING.
387. The Observation for Time, as here described,*
consists in observing the passage, or transit, of a star across
the meridian. The direction of the meridian is supposed to
have been determined by an azimuth observation. If the in-
strument be mounted over a station the azimuth from which
to some other visible point is known, the telescope cah be put
in the plane of the meridian. An observation of the passage
of a star across the meridian will then give the local time, when
the }nean local time of transit of that star has been c )mputed.
In order to eliminate the instrumental errors at least tv/o stars
should be observed, at about the same altitude. If the instru-
ment has no prismatic eye-piece, then only south stars can be
observed with the ordinary field-transits; that is, only stars
having a south declination, if the observer is in about 40° north
latitude. Stars near the pole should not be chosen, since they
move so slowly that a small error in the instrument would
make a very large error in the time of passage.
388. Selection of Stars.— The stars should be chosen in
pairs, each pair being at about the same altitude, or declination.
It is supposed that the American Ephemeris is to be used*
The ** sidereal time of transit, or right ascension of the mean
sun,** is its angle reckoned easterly on the equatorial from the
vernal equinox. This is given in the Ephemeris for every day
of the year. Similarly, the right ascension of many fixed stars
is given for every ten days of the year, under the head of
" Fixed Stars, Apparent Places for the Upper Transit at Wash-
ington." These latter change by a few seconds a year, from
the fact that the origin of coordinates, the vernal equinox itself,
changes by a small amount annually. If, therefore, the hour-
angle, or right ascension, of both the mean sun and a fixed
* It is assumed that the engineer or surveyor has only the ordinary field-
transit, without prismatic eye- piece, so that he can only read altitudes less than
6o*. The accuracy to be attained is about to the nearest sccood of time.
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GEODETIC SURVEYING. 579
star be found for any day of the year, the difference will be the
sidereal interval intervening between their meridian passages,
the one having the greater hour-angle crossing the meridian
much later than the other. When this interval is changed
to mean time the result is the mean or clock time intervening
between their meridian passages. If a fixed star is chosen
whose right ascension is eight hours greater than that of the
mean sun for any day in the year, then this star will come
to the meridian eight hours (sidereal time) after noon, or at
7«» 58'" 41*. 364 after noon of the civil day indicated in the Nau-
tical Almanac. If, therefore, one wishes to make his observa-
tions for time from 8 to 10 o'clock P.M. he should select stars
whose hour-angles, or right ascensions, are from 8 to 10 hours
greater than that of the mean sun for the given date.
In the following table such lists are made out for the first day
of each month for the year 1888. The mean time of transit is
given for the meridian of Washington to the nearest minute, as
well as its mean place for the year. None of these values will
vary more than three or four minutes from year to year, and
therefore the table may be used for any place and for any time.
The table merely enables the observer to select the stars to be
observed. After these are chosen their local mean time of transit
must be worked out with accuracy from the Nautical Almanac*
For any other day of the month we have only to remember that
the star comes to the meridian 3™ 56" earlier (mean time)
each succeeding day, so that for n days after the first of the
month we subtract 3.93 n minutes from the mean time of
transit given in the table, and this will give the approximate
mean time of transit for that date. If we take n days before
♦ Even this trouble may be avoided by using Clarke's Transit Tables (Spon,
London). Price to American purchasers less than one dollar. They arc pub-
lished annually in advance, and give the Greenwich mean time of transit of the
sun and many fixed stars for every day in the year. They are computed for pop
lUaruse from the Nautical Almanac.
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582 SURVEYING.
a date in the table, add 2.93 n minutes to the corresponding
time of transit to find the approximate time of transit for the
given date. This table is therefore a mere matter of conve-
nience to assist in selecting the stars' to be used. They are
nearly all southern stars, since these only can be observed with
the ordinary field-transit.
389. Finding the Mean Time of Transit. — As explained
above, the mean or clock time of transit is simply the sidereal
interval between the mean sun and star for the given place and
date, reduced to mean time. To find this intei*val we find the
right ascension of both mean sun and star, and take their dif-
ference. But the right ascension or sidereal time of the mean
sun or mean noon is given for the meridian of Greenwich,
whereas by the time the sun has reached the given American
meridian its right ascension or sidereal time has increased
somewhat, the hourly increase being 9^.8565. To find the
'* sidereal time of mean noon" for the given place, therefore,
we take the value for the given date for Greenwich and add
to it 9'.8565 for every hour of longitude the place is west of
Greenwich. This then gives the " local sidereal time of mean
noon.** The right ascension of the star, or the sidereal time of
its meridian passage, is then found. This changes only by a
few seconds in a year, and is given for every ten days in the
Washington Ephemeris. This, therefore, needs no correction
to reduce it to its local value for any place. The difference
between the " local sidereal time of mean noon** and the sidereal
time of the star is the sidereal interval of time elapsing between
local mean noon and the transit of the star. When this sidereal
interval is changed to a mean-time interval, which is effected
by means of a table at the back of the Nautical Almanac, the
result is the local mean time of transit of the star.
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- +
59.63
= 19
= 3
42
27
27.43
39.21
= 7
45
I
11.78
16.21
GEODETIC SURVEYING, 583
Example. — Compute the local mean time of transit of 6 Eridani at St. L«uis
on Jan. 16. 1888.
Sidereal lime of mean noon at Greenwich = \^ 41'" 27'.8o
Correction for longitude 6.05'* west
Local sidereal time of mean noon
Right ascension e Eridani Jan. 16
Sidereal interval after mean noon
Correction to reduce to mean time
Local mean time of transit = 7** 43" 55'. 57
390. Finding the Altitude. — The relation between lati-
tude, declination, and altitude is shown by Fig. 146, which rep-
resents a meridian section of the celestial
sphere. Let PP be the line through the
earth's axis ; QQ the plane of the equa-
tor; Z the zenith, and HH' the horizon.
Then H'P=ZQ= <p is the latitude of
the place, and QS=6 and QS'' =^ -6''
are the declinations of 5 and 5" respec-
tively. The altitude of the star 5 is H^Sy ¥ig. 146.
or measured from the south point it would be -^5. The alti-
tude of the star S" is HS'.
We have therefore for altitude of 5
A = //^ - ZG+ 05 = 90^ - 0+ <J.
Also for altitude of 5",
A" =HZ^ ZQ-- QS' = 90" - 0 - d'
But since south declination is considered as negative, we
have, in general, for altitude from the south point, of a star in
the meridian,
^ = 90° - 0 + <y.
The latitude is supposed to be known and the declination
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534 SURVEYING.
is given in the table, whence the altitude of any star in the
list is readily found.
391. Making the Observations. — The meridian is sup-
posed to be established. This may be done either by having
two points in it fixed, one of which is occupied by the instru-
ment and the other by a target, or an azimuth may be known
to any other station or target. In either case the instrument
is put into the meridian by means of both verniers, either mak-
ing the mean of the two read zero on the meridian post, or by
making the mean of their readings on the azimuth station dif-
fer from their mean reading in the meridian by an amount
equal to the azimuth of the given line.
Or, the setting may be approximately on the meridian and
the angle measured so that the true deviation of the instru-
ment from the meridian is observed for each star observation.
The error in time, from a given small error in azimuth, is then
found from the differential equation*
^^sin(0-d)
cos o
where dt is the error in hour-angle in seconds of arc when da
is the deviation from the meridian in seconds of arc, 0 being
the latitude of the place, and 6 the declination of the star.
* One of the fundamental equations that may be written from an inspection
of Fig. II, p. 49. is
cos ^ sin ^ = — cos h sin n,
where h is the altitude and /, ^, and a as above. Diflferentiating with reference
to / and <7, we have
cos h cos a -
dt— da,
cos o cos /
For observations veiy near the meridian both cos a and co« / become
unity, and then we have
cos h sin {0—fi)
af = 9.da =. — da,
cos o cos o
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GEODETIC SURVEYING. 585
Having found the time correction in seconds of arc, the
correction in seconds of time is found by dividing by fifteen.
If the declination is south, or negative, the equation be-
comes
COS o
The error from this cause diminishes as the altitude of the
star increases, and is zero for a zenith observation.
The stars are chosen in pairs, the two stars of a pair hav-
ing about the same altitude or declination. Thus, from the
January group we might select o* Eridani and /? Orionis as
one pair, and fi Eridani and r Orionis for another. The stars
are of course observed in the order of their coming to the
meridian, irrespective of the way they are paired, but they are
paired in the reduction.
The visual angle of the field of view of the ordinary engi-
neer's field-transit is something over one degree. The star
will therefore be visible in the telescope for something over
two minutes before it comes to the vertical wire, it being here
assumed that there is but one vertical thread. Let an attend-
ant hold the watch or chronometer and note the time to the
nearest second when the star is on the wire, as noted by the
observer. If this time be compared with that of the computed
mean time of transit, the error of the chronometer is obtained,
so far as this observation gives it.
The instrument must be reversed on the second star of each
pair. This is to eliminate the instrumental errors. The hori-
zontal angle to the station-mark (whether this be on the
meridian or not) should also be read for every reading on a
star, or at least before and after the star-readings.
The following programme would be adapted to observa-
tions on the four stars selected above :
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5 85 SURVEYING,
PROGRAMME.
1. Set on azimuth station and read horizontal angle (both
verniers).
2. Set in the meridian and read both verniers.
3. Set the approximate altitude of o* Eridani.
4. Note time of passage of o* Eridani.
5. Set on azimuth station and read both verniers.
6. Set in the meridian and read verniers.
7. Note time of passage of ft Eridani.
8. Revolve the telescope 180° on its horizontal axisy relevel^
and read on the azimuth station.
9. Set in the meridian and read verniers.
10. Note time of passage of ft Ononis.
11. Note time of passage of r Orionis.
12. Read both verniers again in the meridian before the
instrument is disturbed.
13. Read to azimuth station.
We have thus obtained four measurements of the hori-
zontal angle, and read with the telescope normal and inverted
on each pair of stars. Especial care must be taken to^see that
the plate-bubble set perpendicular to the telescope is exactly
in the centre when readings are taken to the stars. The mec-in
chronometer error for the two stars of a pair is its true error»
provided it has no rate. If the chronometer has a known rate,
that is, if it is known to be gaining or losing at a certain rate,
then its error must be found for some particular time, as that
of the first observation. Its rate must then be applied to the
observed time of transit of the other stars for the intervening
intervals before comparing results. If local time alone is de-
sired, the result is obtained as soon as a pair of stars has been
observed and their mean result found.
392. Longitude. — If geographical position or longitude is
sought, it remains to compare the chronometer with the
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GEODETIC SURVEYING. 587
Standard or meridian time for that region. This standard time
is now transmitted daily from fixed observatories to almost all
railroad stations in the United States. The time thus trans-
mitted is probably never in error more than a few tenths of a
second. It is usually sent out from 10 A.M. to noon daily. If
the rate of the station clock is known, and also that of the
watch used in the time observation, then a comparison of these
subsequent to the observation would give the difference be-
tween local time and the hourly meridian time used, which
difference changed to longitude would be the longitude of the
place east or west of that standard meridian. If the station
clock cannot be relied on as to its rate, then the watch used in
the observation must have a constant known rate. In this
case the observer compares his watch on the following day
with the time signal as it is transmitted over the railroad com
pany's wires, and so obtains his longitude.
Local time can be observed in this way by means of an ordi
nary transit to the nearest second of time, and the longitude ob
tained to the same accuracy if the rate of the chronometer used
is constant and accurately known. It is probable, however,
that several seconds error may be made if a watch is used,
since probably no watch has a rate which is constant within
one second in twelve hours. Therefore if longitude is desired
a portable chronometer should be used whose rate is well
known.* ^
393. Computing the Geodetic Positions.— After the
angles of the system are adjusted, and the sides of the triangles
computed, we have the plane angles and linear distances from
point to point in the system. It now remains to compute the
* This method has been extensively used for obtaining approximate geodetic
positions for the U. S. Geological Survey in the West, comparisons being made
daily with the Washington University time signals which are transmitted to the
railways in that region.
36
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588 SURVEYING.
4
latitudes and longitudes of the several stations, and the azi-
muths of the lines.
The following formulae, though not exact, are quite suffi-
cient when the sides of the triangles do not exceed ten or
fifteen miles in length : *
NOTATION.
Let L' = latitude of the known point ;
L = latitude of the unknown point;
J/' = longitude of the known point;
M =■ longitude of the unknown point ;
Z^ = azimuth of the unknown point from the known,
counting from the south point in the direction
S.W.N.E.;
Z= azimuth of the known point from the unknown,
or the back azimuth ;
K= length in metres of line joining the two points;
e = eccentricity of the earth's meridian section;
iV= length of the normal, or radius of curvature of a
section perpendicular to the meridian of the
middle latitude, in metres.
R = radius of curvature of the meridian in metres.
Then we have in terms of the length and azimuth of a
given line, in seconds of arc, when the distances are given in
metres, ^
L -L= -dL^ BK cos Z'-\- CK' sin' Z'+ Dh' ;
Z'-i8o°-Z= dZ = dM Sin Z«.
♦ For a summarized derivation of these formula for computing the L M
Z*s from triangulation data, together with extended tables of factors used, see
Report U. S. Coast and Geodetic Survey. 1894, Appendix No. g. The deriva-
tion of the formulae is further amplified in Appendix D of this book.
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GEODETIC SURVEYING,
589
where ^*-^^^_-_; ^«^^,__^; (;«.____.
^. f/sin Z'cosZ'sin i" , 1 ..^ j L + L*
D = ^^—7 o • 2 r>N ; L^ = mean witude = -;
and A = value of first term in right member = BJC cos Z\
Careful attention must be paid to the signs of the Z functions.
TABLE O? LMZ COEFFICIENTS.
Latitude.
Log. y< 4- «>•
Log. ^4-xo.
Log <7+io.
Log. D + 10.
30**
8.5093588
8.5115729
I. 16692
2.3298
31
3363
5054
.18416
.3382
32
3134
4368
.20108
.3460
33
2901
3669
.21772
.3532
34
2665
2959
.23409
.3597
35
8.5092425
8.5112239
I . 25024
2.3656
36
2182
1510
.26617
.3709
37
1936
0772
.28193
.3756
3B
1687
8.5110027
.29753
.3797
39
1437
8.5109275
.31299
.3833
40
8. 5091 I 84
8.5108517
1.32833
3.3863
41
0930
7755
.34358
.3888
42
0675
6989
.35875
•3907
43
0419
6220
.37386
.3921
44
8.5090162
5449
.38894
.3930
45
8.5089904
8.5104677
1.40400
2.3933
46
9647
3905
.41906
.3932
47
9390
3134
.43414
.3924
48
9133
2364
.44926
.3912
49
8878
1598
.46443
.3894
50
8.5088623
8.5100835
1.47968
2.3871
A^l
* /I is to be evaluated for Z. Log sin i" = 4.6855749.
7
J I
\
I 7/.
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S9^ SaiiVEVWG.
Logarithmic values of the coefficients Ay B, C, and D are
given in the above table for each degree of latitude from
30° to 50°. By the aid of this table the LMZ's are readily
found. These tabular values are computed from the constants
of Clarke's spheroid. In this we have
Equatorial semidiameter = 6 378 206 metres.
Polar semidiameter = 6 356 584 **
204.98
Whence the ratio of the semidiameters is -^-^ o-
293.98
Clarke's value of the metre has been taken, which is
I metre = 39.37000 inches.
The difference of azimuth of the two ends of a line is due
to the convergence of the meridians passing through its ex-
tremities, this convergence, as seen from the last of equations
(i), being equal to the difference of longitude into the sine of
the mean latitude.
When the sides of a system of triangulation have been
computed, and the azimuths of the lines are desired from the
several stations, the successive differences of latitude and
longitude are first computed, and from these the azimuths of
the lines, using equations (i). If the longitude is unknown,
the longitude of the first station may be assumed without
affecting the accuracy of the computed relative positions. The
last of equations (l) gives the difference between the forward
and back azimuth of the line joining the two stations. This
difference being applied, with the proper sign, gives the azi-
muth of the first station as seen from the second. But when
the azimuth of one line from a station is known, the azimuths
of all other lines from that station are found from the adjusted
plane-angles at that station, provided the spherical excess h?d
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GEODETIC SURVEYING.
591
been deducted or allowed for, in the adjustment. If no ac-
count has been taken of spherical excess, the error in azimuth
accumulates in working eastward or westward, and soon be-
comes appreciable.
For any other station the azimuth correction is again found
for the line joining this station with a station where azimuths
have been computed, which when applied gives the azimuth of
this line as taken at the forward station, whence the azimuths
of all the lines from this station are known, and so on.
394. Example.— In Fig. 142, p. 5^, let the azimuth of the line CA, from
C, be 80**; latitude of Cbe40°; the length of the line CD be 25000 metres (over
15 miles) ; required the geodetic position of D^ and the azimuth of the line DC
from D,
COMPUTATION OF L M Z,
z
C to A
^.1
00'
48
oo".o
06 .1
A CD — C» (sec 0. 4a*^ - -
39
•"
Zf
dZ
z8o*
Z
C\.qD.
119
48
9
06 .1
49 .5
180
999
38
i6 .6
DtoC
V
dL
L
40*
+
6
oo".ooo
41 .8^7
C
25000 metref .
D
M'
dM
9o»
+
00'
«5
oo".ooo
16 .019
40
06
41 .847
M
90
»S
16 .019
xtttc
tdandadtei
B
K
cosZ'
h
8.5108517
4.3979400
9 6963560
C
sin«Z'
>• 32833
8.79588
9.87679
D
a. 3863
5.aio3
rm
Tns
dL
40" <
- 4o»".853
-1- 1 .006
2.6051477
0.00100
1.0023
759^
0.00395
A
K
sin if'
cos L'(a.c.)
8.5091156
4.3979400
y. 9383948
O.T 164540
dM
SioZa
"dZ
a. 96190
9.80857
-401 .847
J3' aa"
dl
\f
a. 9619044
-f- 9i6".oi9
2.77047
+ S89".48
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592
SURVEYING.
GEODETIC LEVELLING.
395. Geodetic Levelling is of two kinds : {A) TrigonomeU
rical Levelling and {B) Precise Spirit-levelling, In trigonomet-
rical levelling the relative elevations of the triangulat ion-sta-
tions are determined by reading the vertical angles between the
stations, \yhen these are corrected for curvature of the earth's
surface and for refraction it enables the actual difference of
elevation to be found. In precise spirit-levelling a special type
of the ordinary spirit or engineer's level is used, and great
care taken in the running of a line of levels from the sea-coast
inland, connecting directly or indirectly with the triangulation
stations and base-lines. Both these methods will be described.
{A) TRIGONOMETRICAL LEVELLING.
396. Refraction. — If rays of light passed through the atmos-
phere in straight lines, then in trigonometrical levelling we should
have to correct only for the curvature of a level surface at the
locality. It is found, however, that rays of light near the sur-
face of the earth usually are curved downwards — that is, their
paths are convex upwards. This curve is quite variable, some-
times being actually convex downwards in some localities. It
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GEODETIC SURVEYING. 593
has its greatest curvature about daybreak, diminishes rapidly
till 8 A.M., and is nearly constant from lo A.M. till 4 P.M., when
it begins to increase again. The curve may be considered a
circle having a variable radius, the mean value of which is
about seven times the radius of the earth.
397. Formulae for Reciprocal Observations,— In Fig. 14;
the dotted curve represents a sea-level surface.
Let jy= height of station -ff above sea-level;
IT = height of station A above sea-level ;
C = angle subtended by the radii through A and B ;
Z^=. true zenith distance of A as seen from B\
Z = true zenith distance of B as seen from A ;
8 = true altitude of A as seen from B = 90^ — Z ;
6' = true altitude of B as seen from A = 90° — Z' \
h = apparent altitude of A as seen from ^ = d -j- re-
fraction ;
K = apparent altitude of B as seen from ^ = tf' -j- re-
fraction.
d'=- distance at sea-level between A and B\
r = radius of the earth ;
m = coeflficient of refraction.
In the figure join the points A and 5 by a straight line.
This would be the line of sight from -^ to -ff if there were no
refraction. Through A and B draw the radii meeting at C, ex-
tending them beyond the surface.* Take the middle point of
the line AB, as H, and draw HC, Take A A' perpendicular to
HCy and EE' through H and perpendicular to HC. Extend
AA' to meet a perpendicular to it from B. Then do we have
A'C^AC\ EE^AD', and HC^r^^^^'
* lo reality these are the normals at A and B^ but will here be taken as the
radii.
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594 SURVEYING.
Neither of these three relations is quite exact, because
HC does not quite bisect the angle C The figure is greatly
exaggerated as compared to any possible case in practice,
for the angle C would never be more than i° in such work.
The error in practice is inappreciable.
From the geometrical relations shown in the figure we
have
H-H^AB^DBs^z^ (I)
2
But since Cis never more than i°, and usually much less,
we may say
H'-ir =A'B^DB = ADi7iT\BAD.. . . (2)
But AD = E^E— distance between the stations reduced to
their mean elevation above sea-level = rf' ; also
BAD = i{Z-Z^;
.\H-'H' = d'i2in^{Z^Z) (3)
But since d = distance between stations at sea-level, we
have
d \d\\ rH ' : r,
whence we have, for reciprocal observations at A and B,
H-H' = d^^nk{Z-Z){x^^i±P\, . . (5)
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GEODETIC SURVEYING, 59$
or, in terms of S and d',
jy-^' = ^/tanK*'-.<y)(i+-^^), . . (6)
Inhere attention must be paid to the signs of 6 and S\
The effect of refraction is to increase S and 6' by equal
amounts (presumably), whence their difference remains unaf-
fected. Equations (5) and (6) are therefore the true equations
to use for reciprocal observations at two stations. Since the
refraction is so largely dependent on the state of the atmos-
phere, the observations should be made simultaneously for the
best results. This is seldom practicable, however, and therefore
it is highly probable that a material error is made in assuming
that the refraction is the same at the two stations when the
observations are made at different times.
398. Formulae for Observations at One Station only. —
If the vertical angle be read at only one of the two stations,
then the refraction becomes a function in the problem. Since
the curve of the refracted ray is assumed to be circular (it
probably is not when stations have widely different elevations),
the amount of angular curvature on a given line is directly pro-
portional to the length of the line or to the angle C, The dif-
ference 2X A ox B between the directions of the right line AB
and the ray of light passing between them is one half the
total angular curvature of the ray ; that is, it is the angle
between the tangent to the cwved ray at A and the cord AB.
The ratio between this refraction angle at ^ or -5 and the
angle C is a constant for any given refraction curve ; thajt is,
this ratio does not change for different distances between sta-
tions. This ratio is called the coefficient of refraction, and is
Q
here denoted by m. The true angle ^^Z? is equal to <y'-f— ,
but since the observed altitude is increased by the amount of
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396 SURVEYING,
the refraction, we have for the apparent altitude of B, as seen
from A^
C
whence BAD = h''\ mC. (7)
Using this value of the angle BAD in equation (2), we
obtain
= ^;tan(A' + f-«c)(.+^^). . (8)
where h! is positive above and negative below the horizon.
Equation (8) is used where the vertical angle is read from
one station only.
Since the total angular curvature of the ray of light between
A and B is 2mC^ and the curvature of the earth is C, we may
write
Ci 2mC :: r' : r^ or r' = — , • • • (9)
where r' is the radius of the curve of the refracted ray.
Since the curvature of the ray is of the same kind as that
of the earth, but less in amount, the total correction for curva-
C C
ture and refraction is for an angle equal to mC^ -(l— 2w)«
2 2
Also, since C is always a small angle, we may put
C(in seconds of arc) = — -. — -/.
^ r sin 1 '
If the mean radius is used, we have, in feet,
log r r= 7.32020, and log sin i" = 46855749^
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GEODETIC SURVEYING.
597
whence in seconds of arc and distance in feet we have
or
log C = log ^/ — 2.00577
101.34
(10)
or the curvature is approximately equal to \" for 100 feet in
distance.
The following table gives computed values of the combined
mean corrections for curvature and refraction for short dis-
tances, either for horizontal or inclined sights. Both the dis-
tance d and the correction c^ are in feet, except for the last
column, where the distance is given in miles. For a more ex-
tended table for long distances, see page 481.
CORRECTION
FOR EARTH^S CURVATURE
AND REFRACTION.
d
c«
d
c»
d
c.
d
c»
d
i
miles.
c.
3<»
.00a
1300
035
2300
.108
3300
293
! 4300
.379 1
.571
400
.003
1400
040
2400
.118
3400
.237
, 4400
.397 1
a.985
500
.005
1500
046
2500
.128
3500
.951
4500
.415 1
5.142
600
.007
z6oo
052
2600
•>39
3600
.266
4600
.434 1
9.141
700
.010
1700
059
2700
.149
3700
.281
4700
•453
14.28a
800
.013
1800
066
2800
.i6i
3800
.296
4800
.47a
20.567
900
.017
Z900
074
2900
.172
3900
•3"
4900
.49a
27994
zooo
.090
aooo
082
1 3000
.184
4000
.328
5000
.5"
36.563
1 100
.025
a 100
090
3i«>
.197
4100
•345
5100
.533
46.275
laoo
.030
«aoo
099
Saoo
.310
4900
.362
5200
1
•554
10
57.130
399. Formulae for an observed Angle of Depression to
a Sea Horizon. — In Fig. 148 let A be the point of observa-
tion and 5 the point on the sea-level surface where the tangent
from A falls. Then we have
H=AD^*AS\.^ViASD
C
= rtanCtan- (li)
* Let the student prove this relation.
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S98
SURVEYING,
Since the angle C is always very small, we may let the arc
Ar^rrr-r equal its tangent, whence
//'=-tan*C . (12)
2 ^ '
If the observed angle of de-
pression be ^ = C — mC^
then
and
or
C =
I — m'
where h is expressed in seconds of arc.
Log - tan" i" = 6.39032 for distances in feet.
400. To find the Value of m we have
whence
or
or
Z=90°- A + #«C,
Z' = 90° - A' + mC\
Z + Z' ^ 180^ + C= i8o°->4->4' + 2»iC
I — 2;« = —
C '
m
J , h-{-h
■).
(0
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GEODETIC SURVEYING. 599
where h and h are the observed altitudes above the horizon.
It is evident tliat every pair of reciprocal observations at two
stations will give a value for ;//. The mean values of m^ as
found from observations on the United States Coast Survey
in New England, were :
Between primary stations, ..../«=: 0.071
For small elevations, »» = 0.075
For a sea horizon, m =1 0.078
On the New York State Survey the value from 137 obser-
vations was m = 0.073.*
MA- H'
In this work also the term in equations (4) to (8)
never affected the result by more than -^-^ part of its value.
PRECISE SPIRIT-LEVELLING.f
401. Precise Levelling differs from ordinary spirit-level-
ling both in the character of the instruments used and in the
methods of observation and reduction. It is differential
levelling over long lines, the elevations usually being referred
to mean sea-level. In order that the elevations of inland
points, a thousand miles or more from the coast, may be de-
termined with accuracy, the greatest care is required to pre-
vent the accumulation of errors. In order that triangulation
distances may be reduced to sea-level, the elevations of the
bases at least must be found. It is impossible to carry eleva-
tions accurately from one triangulation-station to another by
means of the vertical angles, on account of the great variations
in the refraction. Barometric determinations of heights are
also subject to great uncertainties unless observations be
made for long periods. The only accurate method of finding
the elevations of points in the interior above sea-level is by
* See pages 463 and 464 for a case of excessive refraction profitably utilized.
f See Appendix F for description of methods used on the Miss. River Survey.
See also Trans, Amer, Soc. C, E,, Vols. XXXIX and XLV, for extended papers
and discuss ions. Also Reps, U, S. C, &* G, Survey for 1893, Pt. II, and for
1897, and the An. Reps. Miss. Riv. Com., 1880 to X900.
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600 SURVEYING.
first finding what mean sea-level is at a given point by means
of automatic tide-gauge records for several years, and then
running a line of precise spirit-levels from this gauge inland
and connecting with the points whose elevations are required.
Most European countries have inaugurated such systems of
geodetic leveling, this work being considered an integral part
of the trigonometrical survey of those countries. In the
United States this grade of work was begun on the U. S.
Lake Survey in 1875, by carrying a duplicate line of levels
from a known elevation at Albany, N. Y., and connecting
with each of the Great Lakes.
The Mississippi River Commission, the U. S. Geological
Survey, and the U. S. C. & G. Survey have now (1901) run
several thousand miles of these precise-level lines along the
banks of the Mississippi and Missouri rivers and the Great
Lakes, a transcontinental line along the 39th parallel, and
many lines auxiliary to these.* On all these lines permanent
bench-marks have been set at intervals of from one to five
miles, and their elevations determined above mean sea-level.
402. The Instruments used in this work were at first
those used in Europe for this purpose, but these have been
considerably modified in America in recent years. The style
of the European instrument (made by J. Kern of Aarau,
Switzerland) is shown in Fig. 149. f By turning the head so
as to bring the eyes of the observer into a vertical line, and
observing with both eyes simultaneously, the image of the
bubble, as seen in the mirror, will appear projected upon the
rod, and its stability and position can be noted while reading
the rod. This rod is shown in Fig. 149^. It has no target,
but is divided into centimetre spaces, and is read to milli-
metres by estimation, three horizontal wires being read and
* The author personally conducted some six hundred miles of this work.
Sec map by J. F. Hayford in Trans. Am. Soc. C. E., Vol. XLV, p. 148.
f This instrument made by F. E. Brandis & Sons Co., Brooklyn, N. Y,
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GEODETIC SURVEYING,
6oi
the mean taken. The bubble is placed in the middle of the
scale of the tube, and held there during the reading of the rod
by means of the delicate thumb-screw placed under the eye-
end wye. The bubbles in these instruments are very deli-
cate, one division (2 mm.) on the tube having a value of from
Fig. Z49.
two to four seconds of arc. They are always made with three
leveling-screws, widely spread, and a watch-bubble to assist
in setting. The bubble-tube is chambered at one end so as
to maintain a bubble of the most efficient length at all tem-
peratures. The magnifying power is about 45 diameters.
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6o2
SURVEYING,
Fig. 149a.— Buff and Bbrgbr Prbcisb Lbvbl, No. 2768.
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GEODE TIC S UR VE YING.
602a
Fio. 1493.— Two ViBWS op THS Coast and Gbodbtic Survby Lb?kl op 19001
602^ 5* UR VE YING,
The instrument is always shaded when in service, even the
tripod legs being covered with cloth.
A form of this instrument made by Buff & Berger,
Boston, and designed by Dr. T. C. Mendenhall, then Supt.
U. S. C. & G. Survey, is shown in Fig. 149^, and the
present (1901) form used in that work is shown in Fig. 149^.
This last form of precise level was designed by Mr. J. H.
Hayford, Inspector of Geodetic Work, and Mr. E. G. Fischer,
Chief Mechanician, both of the U. S. C. & G. Survey. In
the language of Mr. Fischer,* ** the aims in designing the
new precise level were to select the material with a view to
the smallness of its expansion coefficient, to protect the vital
parts against sudden and unequal changes of temperature, to
reduce to the smallest possible dimension the linear distance
between level vial and line of collimation, to insure stability
by reducing the distance between the centre of gravity and
the plane of support, and to enable the observer to obtain
the rod-reading, as nearly as possible, simultaneously with the
setting of the level.**
This instrument embodies many novelties. The telescope
is flexibly supported inside a fixed tube, instead of in wyes.
This fixed tube is slotted at top and the level-bubble is
dropped through this into almost immediate contact with the
telescope-tube. All metal parts are made of a nickel-steel
having a small coefficient of expansion, the more important
screws having a coefficient as low as 0.000001 per degree C.f
The bubble-readings are made by means of an upper
mirror throwing the images of the bubble-ends upon two
prismatic reflectors set in an auxiliary tube alongside the tele-
scope, and thence to the left eye of the observer. By stand-
ing erect the observer then sees both the rod and the bubble
* Trans. Am. Soc. C. E., Vol. XLV. p. 128.
f Made by the Soci6t6 Anonyme de Commeiitry-Fourchambault, i6
Place Vend6me, Paris. See foot-note, p. 495.
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GEODETIC SURVEYING,
6o2c
''W
(5 01
■re In oantlmeian
^
$:N
IT— -«<^-H
it **- tr»
Fig. 149^.— Molitor's Precisb-level Rod.
(All dimeosions In centimetres except those on the pin, which arc in inches.)
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6o2^ SURVEYING.
at the same time, and can hold the bubble to a central posi-
tion while reading the rod. A simple lens at the eye end of
this tube (changed for different eyes) brings these images
into the range of distinct vision. This device enables the
bubble to be read simultaneously with the rod, without
parallax, and with a normal or erect position of the body.
It requires a taller tripod, however, than is customary with
other forms of levels. This is probably the most perfect
form of precise level ever devised. It sets very low on the
tripod and has proved very stable in the wind.
Probably the most perfect rod for precise levelling ever
devised is that shown in Fig. 149^.* It is an improvement
upon the Kern rods in several particulars. For short sights
the subdivision of the centimetre spaces would be helpful.
The projecting sides also preserve the face of the rod from
injury. The pin here shown for a turning-point was first
used by the author in 1881. Its bearing-surface is made
convex upward, with a lower groove for sand and dirt which
may be blown in while in use, is important. Foot-plates have
been found less stable than these pins.
In Fig. 149^ is shown the instrumental outfit for a double-
level party when running upon a railroad and using a hand-
car. The tents were used only in strong winds, but the
umbrella shields were used constantly, in even the lightest
winds. Sun-umbrellas were also used at all times, to shade
the instruments. The rod-tripod was used only in adjusting the
rod -bubble every morning, or when removed from its canvas
cover. If a fixed support could be had, like the side of a house,
a fence, or a tree, the rod-tripod could be dispensed with in
adjusting the rod-bubble. This outfit was used by the author
in all his precise-level work referred to in Art. 401. Thirteen
men with a double instrumental outfit rode on this car.
* Designed by Mr. David S. Molitor, M. Am. Soc. C. E., and described
in Trans. Am. Soc. C. E., Vol. XLV, p. 12.
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GEODETIC SURVEYI^.
603
403. The Instrumental Constants which
must be accurately determined once for all, but
re-examined each season, are —
1. The angular value of one division on the
bubble-tube.
2. The inequality in the size of the pivot-
rings.
3. The angular value of the wire-interval, or
the ratio of the intercepted portion on the rod to
the distance of the rod from the instrument.
4. The absolute lengths of the levelling-rods.
These constants may be determined as
follows :
The value of one division of the bubble may
be readily found by sighting the telescope on
the rod, which is set at a known distance from
the instrument, and running the bubble from
end to end of its tube, taking rod-readings
for each position of the bubble. The bubble-
graduations are supposed to be numbered from
the centre towards the ends.
Readings should be taken only for extreme
positions of the bubble, and not for central or
intermediate positions, as those would have little
weight in fixing the average value of one divis-
ion. To test the uniformity of the curvature of
the bubble, however, readings should be taken
for movements of a single division, back and
forth, until many rod-readings have been taken
for each bubble position. Then average rod-
readings could be found for average bubble
positions, differing by about one division only,
and thence the uniformity of the curvature of
the tube determined.
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GEODETIC SURVEYING, 605
Let E, = mean of all the eye-end readings of the bubble
when it was run to the eye-end of its tube ;
E^ = same for bubble at object-end of tube ;
O^ = mean of all the object-end readings when bubble
was at eye-end of tube ;
(?, = same for bubble at object-end of tube;
R^ = mean reading of rod for bubble at eye-end ,
i?, = same for bubble at object-end ;
D = distance from instrument to rod ;
V = value of one division of the bubble (sine of the
angle) at a unit's distance.
In seconds of arc we would have
z; (in seconds) = /^TJ^' e^_ qS ' ^'^
D sin I ' \^ ^ ^1
If a table is to be prepared for corrections to the rod-read-
ings for various distances and deviations of the bubble from
the centre of its tube, then the value as given by equation (i)
is most convenient to use. The value of one division of a level
bubble should be constant, but it is often affected by its rigid
fastenings, which change their form from changes in tempera-
ture.
The inequality in the size of the rings is found by revers-
ing the bubble on the rings, and also reversing the telescope
in the wyes. The bubble is reversed only in order to eliminate
its error of adjustment. The following will illustrate the
method of making and reducing the observations :
37
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6o6
SURVEYIXG.
BUBRLB-READINCS
North.
South.
Tel. eye end north.
Lev. direct.
4.3
5.5
<( «t (<
'* reversed.
4.7
9.0
(-1.7)
— 0.42
5.2
10.7
Tel. eye-end south
Lev. direct.
6.2
3.7
<< n .<
*' reversed.
6.6
12.8
(+5.8)
3.3
7.0
Tel. eye- end north
Lev. direct
4.4
+ 1.45
5-5
" " "
*' reversed.
4.8
9.2
(-1.5)
5.2
10 7
Mean reading north
= — 0.40
— 0.38
south
= + 1.45
North minus south = — 1.85
That is to say, the bubble moves 1.85 divisions towards the
object-end when the telescope is reversed in the wyes. This is
evidently twice the inequality of the pivot-rings ; and since the
axis of a cone is inclined to one of its elements by one half
the angle at the apex, so the line of sight is inclined to the
tops of the rings by one fourth of 1.85 divisions, or 0.46 divi-
sions of the bubble. It is also evident that the eye-end ring
is the smaller, and that therefore when the top surfaces of the
rings are horizontal the line of sight inclines downward from
the instrument. The correction is therefore positive. This is
called the pivot-correction, and changes only with an unequal
wear in the pivot-rings.
The angular value of t lie wire-interval \s found by measur-
ing a base on level ground of about 300 feet from an initial
point f* in front of the objective. Focus the telescope on
a very distant object, and measure the distance from the
inside of the objective to the cross-wires, this being the value
* See art. 20Q for the significance of this terra, as well as for the theory of the
pvobiem.
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GEODETIC SURVEYING, 607
of /for that instrument. Measure the space intercepted on
the rod between the extreme cross-wires.
If d =: length of base, counting from the initial point;
s = length of the intercepted portion of the rod ;
/
r = T- = constant ratio of distance to intercept ;
d
then r = - ;
s
and for any other intercept / on the rod we have
rf' = r/+/+c (3)
When r,/, and c are found, a table can be prepared giving
distances in terms of the wire-intervals.
The errors in the absolute lengths of the rods affect only
the final differences of elevation between bench-marks. This
correction is usually inappreciable for moderate heights.
404. The Daily Adjustments. — The adjustments which
are examined at the beginning and close of each day's work
are as follows :
1. The collimation, that is, the amount by which the Hne
of sight, as determined by the mean reading of the three wires,
deviates from the line joining the centres of the rings.
2. The bubble-adjustment — that is, the inclination of the
axis of the bubble to the top surface of the rings.
3. The rod-level. This is examined only at the beginning
of each day's work, and made sufficiently perfect.
The first two adjustments are very important, since it is by
means of these (in conjunction with the pivot-correction,
determined once for the season) that the relation of the bubbl*!
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6o8 SURVEYIXG.
to the line of sight is found. It is not customary in this work
to try to reduce these errors to zero, but to make them reason-
ably small, and then determine their values and correct for
them. It is evident that if the back and fore sights be kept
exactly equal between bench-marks, then the errors in the
instrumental adjustments are fully eliminated ; and in any case
these errors can only affect the excess in length of the sum of
tlie one over that of the other. It is to this excess in length
of back-sights over fore-sights, or vice versa^ that the instru-
mental constants are applied ; but in order to apply them their
values must be accurately determined. The curvature of a
level surface would also enter into this excess, but it is usually
so small a residual distance, that the correction for curvature
is quite insignificant. There are, however, three instrumental
corrections to be applied for the amount of the excess, namely,
the corrections for coUimation, inclination of bubble, and in-
equality of pivots, designated respectively by ^, i, and/. Since
three horizontal wires are read on the rod, the wire-intervals
can be used in place of the distances, for they are linear func-
tions practically, and so a record is kept of the continued sum
of the lengths of the back and fore sights, and from these the
final difference is found.
The collimation-correction is taken out for a distance of
one unit (the metre has been universally used in this kind of
levelling), and then the correction for any given case found by
multiplying by the residual distance.
Let R^ = rod-reading for telescope normal ;
i?,= '' " " " inverted;
d = distance of rod from instrument.
Then ' = ^^ (0
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GEODETIC SURVEYING, 609
The correction for the inclination of the bubble to the tops
of the rings is found by reversing the bubble on the telescope
and reading it in both positions. In such observations the
initial and final readings are taken with the bubble in the same
position, thus giving an odd nunnber of observations. Usually
two direct and one reversed reading are taken. The correction
is found in terms of divisions on the bubble, the correction in
elevation being taken from the table prepared for that purpose.
Let E^ = fpuan of the eye-end* readings for level direct ;
E^ = ** " " " " ** reversed ;
O, = " " object " " " direct ;
(?, = " " " " *' " reversed;
then
L(&^_&^) (.)
The pivot correction has already been found, and is sup-
posed to remain constant for the season.
If E be the excess of the sum of the back-sights over that
of the fore-sights, then the final correction for this excess is
C=£[^ + K^-+/)], (3)
where v is taken from eq. (i), p. 581. Evidently, if the fore-
sights are in excess, the correction is of the opposite sign.
405. Field Methods. — The great accuracy attained in pre-
cise levelling is due quite as much to the methods used and
precautions taken in making the observations as to the instru-
mental means employed. Aside from errors of observation
and instrumental errors, we have two other general classes of
* By eye-end is always meant the end towards the eye end of the telescope,
whether in a direct or a reversed position.
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-lO SURVEYING,
' errors, which can be avoided only by proper care being used
in doing the work. These two classes are errors from unstable
supports and atmospheric errors.
Any settling of the rod between the fore and back readings
upon it will result in the final elevation being too high, while
^any settling of the instrument between the back and fore
readings from it will also result in too high a final elevation.
Such errors are therefore cumulative, and the only way in
which they can be eliminated is to duplicate the work over
the same ground in the opposite directionr. As a general pre-
caution, the duplicate line should always be run in the opposite
direction. This will result in larger discrepancies than if both
are run in the same direction, but the mean is nearer the truth.
Atmospheric errors may come from wind, heated air-cur-
rents causing the object sighted to tremble or "dance," or
from variable refraction. For moderate winds the instrument
may be shielded by a screen or tent, but if its velocity is more
than eight or ten miles an hour, work must be abandoned.
To avoid the evil effects of an unsteady atmosphere the length
of the sights is shortened ; but when a reading cannot be well
taken at a distance of about 150 feet, or 50 metres, it would
be better to stop, since the errors arising from the number of
stations occupied would make the work poor. At about 8
o'clock A.M. and 4 P.M. very large changes in the refraction
have been observed on lines over ground which is passing from
sun to shade, or vice versa, when the image was apparently
very steady. In clear weather not more than three or four^
hours a day can be utilized for the best work, and sometimes,
with hot days and cool nights, it is impossible to get an hour
when good work can be done.
In making the observations the bubble is brought exactly
to the centre of its tube, the observer being able to do this
by means of the thumbscrew under one wye, and the mirror
which reflects the image of the bubble to the observer at the
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GEODETIC SURVEYING. 6ll
eye-piece. If there is no mirror to the bubble, then it is
brought approximately to the centre, and the recorder reads
it while the observer is reading the three horizontal wires. In
any case the bubble-reading is recorded in the note-book, and
if it was not in the middle a correction is made for the eccen-
tric position by means of a table prepared for the purpose.
The mean of the three wire-readings is taken as the reading
of that rod, the observer estimating the tenths of the centi-
metre spaces, thus reading each wire to the nearest millimetre.
The wires should be about equally spaced so that the mean of
the three wires coincides very nearly with the middle wire.
The differences between the middle and extreme wire-readings
are also taken out to give the distance, as well as to check the
readings themselves by noting the relation of the two intervals.
If they are not about equal, then one or more of the three
readings is erroneous. This is a most important check, and
constitutes an essential feature of the method.
It has been found economical to have two rodmen to each
instrument, so that no time shall be lost between the back and
fore sight readings from an instrument-station. Since but a
small portion of the day can generally be utilized, it is desira-
ble to make very rapid progress when the weather is favora-
ble. When two rodmen are used, and the air is so steady that
loo-metre sights can be taken, it is not diflficult for an expe-
rienced observer to move at the rate of a mile an hour.
On the U. S. Coast and Geodetic Survey* a much more
laborious method of observing than the one above outlined
has been followed. There a special kind of target-rod has
been employed, the target being set approximately and
clamped. The thumb-screw under the wye is used as a mi-
crometer-screw, and two readings are taken on it, one when
*The elaborate methods of observing here described were found in 1899 to
give some seven times the absolute errors involved in the methods described in
the fore part of this article, and as practiced by the author, and hence they were
abandoned in favor of the simpler and more rapid methods. In the final adjust-
ment of all the systems the work done under the U. S. A. Engr. Corps by the
author and others was f^iven a weight ninety.five times as great as that done
previous to 1899 by the U. S. C. h G. Survey. See Rep. U. S. C ^ G. Survey^
1S98-9, Appendix 8, pp. 433-3. ( \
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J
6l2 SURVEYING.
the bubble is in the middle and the other when the centre
wire bisects the target, the bubble now not being in the
middle, since the target's position was only approximate. The
bubble is then reversed, and two more readings of the screw
taken. The telescope is now revolved in the wyes, and read-
ings taken again with bubble direct and reversed. Thus there
are four independent readings taken on the rod, each necessi-
tating two micrometer-readings. The reduction is also very
complicated, each sight being corrected for curvature and re-
fraction as well as for instrumental constants. The duplicate
line is carried along with the first one by having two sets of
turning-points for each instrument-station. The instrument,
however, is set but once, so that the lines are not wholly inde-
pendent. The alternate sections are run in opposite directions;
thus partly obviating the objection to nmning both lines in
the same direction. The method first described was used on
the U. S. Lake and Mississippi River surveys, and is also the
method used on most of the European surveys of this char-
acter.
The instrument is always shaded from the sun, both while
standing and while being carried between stations. It is abso-
lutely necessary to do this in order to keep the adjustments ap-
proximately constant, and the bubble from continually moving.
" - 406. Limits of Error. — On the U. S. Coast and Geodetic
Survey the limit of discrepancy between duplicate lines is
4""" ^ K* where K is the distance in kilometres. On the U. S.
Lake Survey the limit was_jo"*" ^ K, and on the Mississippi
River Survey it was 5™" ^ K. These limits are respectively
0.017, 0.041, and 0.021 feet into the square root of the distance
in miles. If any discrepancies occurred greater than these the
stretch had to be run again.
The " probable error " of the mean of several observations
on the same quantity is a function of the discrepancies of the
* When the old method of observing was used on this service the limit was
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GEODETIC SURVEYING, 613
several results from the mean. If v^y z\, z/„ etc., be the several
residuals obtained by subtracting the several results from the
mean, and if -2'[z/z/] be the sum of the squares of these residu-
als, and if m be the number of observations, then t\\Q probable
error of the mean is i? = ± .6745
\l m{m — i)
This is the function which is universally adopted for meas-
uring the relative accuracy of different sets of observations.
If there be but two observations this formula reduces to
where V\s the discrepancy between two results.
The European International Geodetic Association have
fixed on the following limits of probable error per kilometre
in the mean or adopted result: ±3'"'" per km. is tolerable;
± 5""" per km. is too large ; ± 2™*" per km. is fair; and ± i'"'"
per km. is a very high degree of precision. On the U. S.
coast and geodetic line from Sandy Hook to St. Louis, a dis-
tance of 1 109 miles, the probable error per kilometre was
± 1.2°"".* For the 670 miles of this work on the Mississippi
River Survey, of which the author had charge, the probable
error of the mean for the entire distance was 23.5'""' (less than
one inch), and the probable error per kilometre was ± o.7"'".t
Of course very little can be predicated on these results as to the
actual errors of the work, since the number of observations on
each value was usually but two ; but they may fairly be used
for the purpose of comparing the relative accuracy of different
lines where this function has been computed from similar
data.
407, Adjustment of Polygonal Systems in Levelling.— If
* Report U. S. Coast and Geodetic Survey, 1882, p. 522.
f Reports of the Mhs. Riv. Commission for the years 1882, 1883, and 1884
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6l4 SURVEYING,
a line of levels closes upon itself the summation of all the differ-
ences of elevation between successive benches should be zero.
If it is not, the residual error must be distributed among the
several sides, or stretches, composing the polygon, according
to some law, so that the final corrections which are applied to
the several sides shall be independent of all personal considera-
tions. These corrections should also be the most probable
corrections. There are two general criterions on which to
found a theory of probabilities. One may be called a priori^
and the other a posteriori. By the former we would say that
the errors made are some function of the distance run, as that
they are directly proportional to this distance, or to the square
root of this distance, etc.; while by the latter, or a posteriori
method, we would say the errors made on the several lines are
a function of the discrepancies found between the duplicate
measurements on those lines, or to the computed ** probable
error per kilometre," as found from these discrepancies. Both
methods are largely used in the adjustment of observations.
These laws of distribution are equivalent to establishing a
method of weighting the several sides of the system, a larger
^jj^eight implying that a larger share of the total error is to be
given to that side. When any system of weights is fixed upon,
then the corrections may be computed by the methods of least
squares so as to comply with the condition that the corrections
shall be the most probable ones for that system of weighting.*
The most probable set of corrections is that set the sum of
whose squares is a minimum. If the system includes more
than a few polygons, this method of reduction is exceedingly
laborious, while the increased accuracy is very small over that
from a much simpler method.
Fig. 150 represents the Bavarian network of geodetic levels,
there being four polygons. Every side has been levelled, and
the difference of elevation of its extremities found. These ele-
vations must now be adjusted so that the differences of eleva-
* The word ** weight " as used above has the meaning it has on p. 200, but
it is just the opposite to the meaning of the word in "Least Square" work,
where greater weight means greater accuracy.
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GEODETIC SVRVEYIXG.
615
Fig. 150.
tion on each polygon shall sum up zero. When these sums
are taken the following residuals are found : I., + 20.2™"; II.,
+ 39.3"™; III., — 25.2"""; and
IV., + 108.0™"'. It was sup-
posed that an error of one deci-
metre had been made in the
fourth polygon, but in the ab-
sence of any knowledge in the
case this error must be distrib-
uted with the rest.
The method which the au-
thor would recommend is a
modification of Bauernfeind's,
in that the errors are to be made
proportional to the square roots of the lengths of the sides in-
stead of the lengths of the sides directly. Since the errors in
levelling are compensating in their nature they would be ex-
pected to increase with the square root of the length of the
line, and it is the author's experience that the error is much
nearer proportional to the square root of the distance than to
the distance itself.
Instead of treating the four polygons as one system and
solving by least squares, the polygon having the largest error
of closure \% first adjusted by distributing the error among its
sides in proportion to the square roots of the lengths of those
sides. Then the polygon having the next largest error is ad-
justed, using the new value for the adjusted side, if it is con-
tiguous to the former one, and distributing the remaining
error among the remaining sides of the figure, leaving the
previously adjusted side undisturbed. The adjustment pro-
ceeds in this manner until all the polygons are adjusted. The
Bavarian system is worked out on this plan in the following
tabulated form :
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6i6
PURVEYING.
^\^^
ADJUSTMENT OF THE BAVARIAN SYSTEM OF LEVEL
POLYGONS.
No.
Si.le.
Lensfth.
Sq. Root
of
Length
= A.
No.
Polygon.
2A.
Difference
of
Elevation.
Error
of
Closure
Cor-
rected
Error
ot
Closure
Cor-
rcciion.
Corrected
Difference
of
Elevation.
km.
m.
mm.
I
125.8
II. 2
I.
24.6
+ 35.8723
+ 20.2
+ 31.3
- "43
+ 35.8580
3
179.0
13 4
I.
- 217.5062
- 17.0
- 217.523*
3
M7-3
12. 1
II.
± 181.6541
+ 39-3
+ 39-3
— II. I
± 181.6652
4
60.6
7.8
II.
43- 1
+ 3»095&
- 7.1
+ 320887
5
174.0
13.2
II.
+ 179-5981
— 12.0
+ 179.5861
6
lOI.I
10. 0
II.
20.9
T 30.0005
- 25.2
+ "9-9
- 9.1
T 30.0096
7
J34-9
11.6
III.
- 38.6644
— 11.0
- 386754
8
80.1
9.0
IV.
T 48.8053
-36.0
± 48.7<^3
9
87.0
9-3
III.
+ 57 4440
- 8.9
+ 57-435i
ID
96.8
9.8
IV.
27.0
— 100.1619
-f- 108.0
+ 108.0
-39.«
— 100.201I
II
67.9
8. a
IV.
+ 5t.4646
-3».8
+ 5".43«8
Beginning with polygon IV., we find its error of closure to
be + 108.0*""', this being distributed among the three sides so
that -^ goes to side 8, -^j^ to side 10, and t^ to side il.
The corrected values for these sides are now found. Next
take the polygon having the next largest error of closure,
which is number II., and distribute its error in like manner.
This leaves polygons I. and III. to be adjusted, one side of
the former and two of the latter being already adjusted. The
corrected errors of closure for these polygons are 31.3""* and
19.9""" respectively, the former to be di.stributed between the
sides I and 2 and the latter between the sides 7 and 9. The
resulting corrected values cause all the polygons to sum up
zero.
The sum of the squares of the corrections here found is
50.02 square centimetres, whereas if the differences of eleva-
tion had been weighted in proportion to the lengths of the
sides and the system adjusted rigidly by least squares the sum
of the squares of the corrections would have been 49.65 square
centimetres, showing that the method here used is practically
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GEODETIC SURVEYING. 617
as good as the rigid method which is commonly used. It has
been found in practice to give, in general, about the same
sized corrections as the rigid system.
408. Determination of the Eleyation of Mean Tide.—
To determine accurately the elevation of mean tide at any
point on the coast requires continuous observations by means
of an automatic self-registering gauge for a period of several
years. The methods of making these observations with cuts
of the instruments employed are given in Appendix No. 8 of
the U. S. Coast Survey Report for 1876. A float, inclosed in
a perforated box, rises and falls with the tide, and this motion,
properly reduced in scale by appropriate mechanism, is re-
corded by a pencil on a continuous roll of paper which is moved
over a drum at a uniform rate by means of clockwork. An
outer staff-gauge is read one or more times a day by the at-
tendant, who records the height of the water and the time of
the observation on the continuous roll. This outer staff is
connected with fixed bench-marks in the locality by very
careful levelling, and this connection is repeated at intervals to
test the stability of the gauge.
To find from this automatic record the height of mean tide,
ordinates are measured from the datum-line of the sheet to
the tide-curve for each hour of the day throughout the entire
period. This period should be a certain number of entire
lunar months. The mean of all the hourly readings for the
period maybe taken as mean tide. It maybe found advisable
to reject all readings in stormy weather, in which case the
entire lunation should be rejected.
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CHAPTER XV.
PROJECTION OF MAPS, MAP- LETTERING, AND TOPO-
GRAPHICAL SYMBOLS.
L PROJECTION OF MAPS.
409. The particular method that should be employed in
representing portions of the earth's surface on a plane sheet
or map depends, first, on the extent of the region to be repre-
sented ; second, on the use to be made of the map or chart ;
and thirds on the degree of accuracy desired.
Thus, a given kind of projection may suffice for a small
region, but the approximation may become too inaccurate
when extended over a large area. It is quite impossible to
represent a spherical surface on a plane without sacrificing
something in the accuracy of the relative distances, courses,
or areas ; and the use to which the chart is to be put must de-
termine which of these conditions should be fulfilled at the
expense of the others. A great many methods have been
proposed and used for accomplishing various ends, some of
which will be described.
410. Rectang^ular Projection.— In this method the merid-
ians are all drawn as straight parallel lines ; and the parallels
are also straight, and at right angles with the meridians. A
central meridian is drawn, and divided into minutes of latitude
according to the value of these at that latitude as given in
Table VIII. Through these points of division draw the paral-
lels of latitude as right lines perpendicular to the central
meridian. On the central parallel lay off the minutes of
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PROJECTION OF MAPS, 619
longitude, according to their value for the given latitude, by
Table VIII. ; and through these points of division draw the
other meridians parallel with the first.
The largest error here is -in assuming the meridians to be
parallel. For the latitude of 40°, two meridians a mile apart
will converge at the rate of about a foot per mile. A knowl
edge of this fact will enable the draughtsman to decide when
this method is sufficiently accurate for his purpose. Thus, for
an area of ten miles square, the distortion at the extreme cor-
ners in longitude, with reference to the centre of the map as
an origin of coordinates, will be about twenty-five feet. At
the equator this method is strictly correct.
In this kind of projection, whether plotted from polar or
rectangular coordinates, or from latitudes and longitudes, all
straight lines of the survey, whether determined by triangula-
tion or run out by a transit on the ground, will be straight on
the map ; that is, the fore and back azimuth of a line is the
same, or, in other words, a straight line on the drawing gives
a constant angle with all the meridians.
This is the method to use on field-sheets, where the survey
has all been referred to a single meridian.
411. Trapezoidal Projection.— Here the meridians are
made to converge properly, but both they and the parallels
are straight lines. A central meridian is first drawn, and grad-
uated to degrees or minutes ; and through these points paral-
lels are drawn, as before. Two of these parallels are selected ;
one about one fourth the height of the map from the bottom,
and the other the same distance from the top. These paral-
lels are then subdivided, according to their respective lati-
tudes, from Table VIII. ; and through the corresponding points
of division the remaining meridians are drawn as straight lines.
The map is thus divided into a series of trapezoids. The
parallels are perpendicular to but one of the meridians. The
principal distortion comes from the parallels being drawn at
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620 SURVEYING.
straight lines, and amounts to about thirty-two feet in ten
miles in latitude 40°, and is nearly proportional to the square
of the distance east or west from the central meridian.
The work should be plotted from computed latitudes and
longitudes. The method is adapted to a scheme which has a
system of triangulation for its basis, the geodetic position of
the stations having been determined. These conditions would
be fulfilled in a State topographical or geological survey for
the separate sheets, each sheet covering an area of not more
than twenty-five miles square.
412. The Simple Conic I^/ojention.— In this projection,
points on a spherical surface are first projected upon the sur-
face of a tangent cone, and then this conical surface is devel-
oped into the plane of the map. The apex of the cone is
taken in the extended axis of the earth, at such an altitude
that the cone becomes tangent to the earth's surface at the
middle parallel of the map. When this conical surface is de-
veloped into a plane, the meridians are straight lines converg-
ing to the apex of the cone, and the parallels are arcs of con-
centric circles about the apex as the common centre.
The sheet is laid out as follows: Draw a central meridian,
and graduate it to degrees or minutes, according to their true
values as given in Table VIII. Through these points of divi-
sion draw parallel circular arcs, using the apex of the cone as
the common centre. For values of the length of the side of
the tangent cone, which is the length of the central parallel
above, see Table VIII. The rectangular coordinates of points
in these curves are also given in the same table.
On the middle parallel of the map the degrees or minutes
of longitude are laid off, and through these are drawn the re-
maining meridians as straight lines radiating from the apex
of the tangent cone.
It will be seen that the latitudes are correctly laid off, and
the degrees of longitude will be sufficiently accurate for a map
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PROJECTION OF MAPS, 62 1
covering an area of several hundred miles square. The merid-
ians and parallels are at right angles.
In this projection the degrees of longitude on all parallels,
except the middle one; are too great; and therefore the area
represented on the map is greater than the corresponding area
on the sphere.
The chart should be plotted from computed latitudes and
longitudes.
413. De I'Isle's Conic Projection. — This is very similar
to the above, except that two parallels, one at one fourth, and
one at three fourths the height of the map, are properly grad-
uated, and the meridians drawn as straight lines through these
points of division. The parallels are drawn as concentric cir-
cles, as in the simple conic projection. This is therefore but a
combination of the second and third methods, and is more
accurate than either of them. The cone here is no longer tan-
gent, but intersects the sphere in the two graduated parallels.
In this case the region between the parallels of intersection is
shown too small, and that outside these lines is shown too
large ; so that the area of the whole map will correspond very
closely to the corresponding area on the sphere. When these
parallels are so selected that these areas will be to each other
exactly as the scale of the drawing, then it is called " Mur-
doch's projection."
414. Bonne's Projection. — This differs from the simple
conic in this — that all the parallels are properly graduated,
and the meridians drawn to connect the corresponding points
of division in the parallels. These latter are, however, still
concentric circles. The meridians arc at right angles to the
parallels only in the middle portion of the map. The same
scale applies to all parts of the chart. There is a slight dis-
tortion at the extreme corners, from the parallels being arcs
of concentric circles. The proportionate equality of areas is
38
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622 SURVEYING,
preserved. A rhumb-line appears as a curve ; but when once
drawn, its length may be properly scaled.
It will be noted that the last three methods involve the
use of but one tangent or intersecting cone.
415. The Polyconic Projection. — For very large areas it
is preferable to have each parallel the development of the
base of a cone tangent in the plane of the given parallel.
This projection differs from Bonne's only in the fact that the
parallels are no longer concentric arcs, but each is drawn with
a radius equal to the side of the cone which is tangent at
that latitude. These, of course, decrease towards the pole ;
and therefore the parallels diverge from each other towards
the edge of the chart. The result of this is, that a degree
of latitude at the side of the map is not equal to a degree
on the central meridian ; or, in other words, the same scale
cannot be applied to all parts of the map. These defects ap-
pear, however, only on maps representing very large areas.
The whole of North America could be represented by this
method without any material distortion.
This method of projection is exclusively used on the Unit-
ed States Coast and Geodetic Survey, and for all other maps
and charts of large areas in this country. Extensive tables are
published by the War and Navy Departments, and also by
the Coast Survey, to facilitate the projection of maps by the
polyconic system. Table VIII. gives in a condensed form the
rectangular coordinates of the points of intersection of the
parallels and meridians referred to the intersection of the sev-
eral parallels with the central meridian as the several origins.
416. Formulae used in the Projection of Maps.* — The
fundamental relations on which the method of polyconic pro-
jection rests are given in the following formulae:
* See Appendix D for the derivation of equations (i) and (2).
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PROJECTION OF MAPS, 623
Normal, being the radius of curvature
of a section perpendicular to the y,
meridian at a given point iV= _'_—-, (i)
where R, is the equational radius,
e is the eccentricity,
and L is the latitude.
Radius of the meridian Rm — N""^^ ^ ,— . . (2)
Radius of the parallel Rp -^ N cos L. . . (3)
Degree of the meridian D„, - ~~Q~^^n ... 1^4)
= 3600^?,^ sin i''.
Degree of the parallel ^/ = "o~ ^/ • • • (5)
= 3600-^/ sin i".
Radius of the developed parallel, or
side of tangent cone r =• N cot L. . , . (6)
If n be any arc of a parallel, in' degrees, or any difference
of longitude from the central meridian of the drawing, and
if B be the corresponding angle, in degrees, at the vertex of
the tangent cone, subtended by the developed parallel, then
since the angular value of arcs of given lengths are inversely
as their radii, we have
-= -^ = sin L.
n r '
or 0 — n sin L (8)
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624 SU/^VKYJ\C.
Since the developed parallels are circular arcs, the rectangu^
lar coordinates of any point an angular distance of 6 from
the central meridian is,
Meridian distance, rf^ = ;r = r sin d.
Divergence of parallels, d^ = y =: r vers 0. V . , (9)
= X tan ^0'
For arcs of small extent, the parallel may be considered
coincident with its chord ; but the angle between the axis of x
and the choid is \d. If, then, the length of the arc, which is
nDp, be represented by the chord, we may write
df„ = meridian distance z= x = nDp cos i^A / \
and dp = divergence of parallels = ^ =; uD^ sin -Jft f
If, now, dmx = meridian distance for i degree of longitude,
and d^n = meridian distance for n degrees of longitude,
d^n nDp cos \e^
we have i~ = 77- rs-.
d^^ • Dp cos \t)^
But 6^11 sin Z, so that ^, = i*^ X sin L = 38' for latitude 40^
Therefore
cos \B^ = cos 19' = I, nearly;
so that ^ = n cos \{n sin L), nearly (n)
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PROJECTION OF MAPS. 625
For L = 30°, we have sin Z = J. Therefore, for latitude 30**,
^^ = n cos in= n cos (o.25«), nearly.
If we have obtained the meridian distance, d^, for i degree
of longitude, and wish to obtain it for n degrees in latitude
30°, we have but to multiply the distance for i degree by n
cos (0.25;/).
417. In Table VIII. the meridian distances are given, at vari-
ous latitudes, for a difference of longitude of one degree. To
find the meridian distance for an)-^ number of degrees or parts
of degrees, multiply the distance for one degree by the factor
there given for the given latitude. The factor given in the
table for latitude 30° is n cos (0.288;?), in place of n cos (o.25«),
as obtained above. The difference is a correction which has
been introduced to compensate the error made in assuming
that the chord was equal in length to its arc. The corrected
factors enable the table to be used without material error up
to 25 degrees longitude either side of the central meridian.
To obtain the divergence of the parallels for differences of
longitude more or less than one degree, multiply the diver-
gence for one degree by the square.of the number of degrees.
It is evident that this rule is based on ^he assumption that the
arc of the developed parallel is a parabola, and so it may be
considered for a distance of 25 degrees either side of the cen-
tral meridian between the latitudes 30° and 50° without mate-
rial error.
If the whole of the United States were projected by this
table, using the factors given, to a scale of i to 1,500,000, thus
giving a map some 8 by 10 feet, the maximum deviation of
the meridians and parallels from their true positions (which
would be at the upper corners) would be but about 0.02 inch.
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SURVEYING,
Thus, for a map of this size, covering 20 degrees of lati-
tude and 50 degrees of longitude, the geodetic lines would
have their true position within the
width of a fine pencil line, by the use
of Table VIII. Fig. 151 will illus.
trate the use of the table in project-
ing a map by the polyconic method.
The map covers 30 degrees in lati-
tude (30° to 60°) and 60 degrees in
longitude. The straight line OJD^ is
first drawn through the centre of the map, and graduated ac-
cording to the lengths of one degree of latitude, as given in
the second column of Table VIII. Through these points of di-
vision the lines m^ m^, are drawn in pencil at right angles to
the central meridian. On these lines the points nti, m^^ etc.,
are laid off by the aid of the first part of Table VIII. This ta-
ble gives the meridian distances when n is less than one degree,
as well as when n is greater. From the points mi, Wj, etc.^
the divergence of the parallels is laid off above the lines OfHy
by the aid of the second portion of Table VIII., thus obtaining
the positions of the points/,. /j, etc. The points/ mark the
intersection of the meridians and parallels ; and these may
be drawn as straight lines between these points, provided a
sufficient nuipber of sucji points have been located. The map
is then to be plotted upon the chart from computed latitudes
and longitudes.
418. Summary. — We have seen that there are, in general,
two ways of plotting a map or chart, and two corresponcing
uses to which it may be put:
First. We may plot by a system of plane coordinates,
either polar (azimuth and distance) or rectangular (latitudes
and departures). This gives a map from which distance,
azimuth (referred to the meridian of the map), and areas are
correctly determined.
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PROJECTION OF MAPS, 627
Second. We may plot the map by computed latitudes and
longitudes, and determine from it the relative position of points
in terms of their latitude and longitude.
The first system is adapted to small field sheets and detail
charts for which the notes were taken by referring all points
to a single point and meridian. For this purpose the system
of rectangular projection should be selected, as long as the
area of the chart is not more than about one hundred square
miles. If it be larger than this, the trapezoidal system should
be used. In case this is done, the work is still plotted as
before, provided it has all been referred to a given meridian in
the field work, and then converging meridians are drawn as
described above. From such a chart, not only the azimuth
(referred to the central meridian) and distance may be deter-
mined, but the correct longitude and nearly correct latitude
are given.
In the case of topographical charts, based on a system of
triangulation, each sheet is referred to a meridian passing
through a triangulation-station on that sheet, or near to it,
and the rectangular system used.
In the case of a survey of a long and narrow belt, as
for a river, railroad, or canal, if the survey was based on a
system of triangulation, the convergence of meridians has been
looked after in the computation of the geodetic positions of
these stations, and each sheet is plotted by the rectangular
system, being referred to the meridian through the adjacent
triangulation-station. When many of these are combined into
a single map on a small scale, then they must be plotted on
the condensed map by latitudes and longitudes, these being
taken from the small rectangular projections, and plotted on
the reduced chart in polyconic projection ; the meridians and
parallels having been laid out a.<^ shown above.
In case the belt extends mostly east and west, and is not
based on a triangulation scheme, then observations for azimuth
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628 SURVEYING.
should be made as often as every fifty miles. It will not do
to run on a given azimuth for this distance, however; for there
has been a change in the direction of the parallel (or meridian)
in this distance, in latitude 40*^, of about 40 minutes. Accord-
ing to the accuracy with which the work is done, therefore,
when running wholly by back azimuths, the setting of the in,
strument must be changed. Thus, if in going i degree (53
miles), east or west, in latitude 40^, the meridian has shifted
40', then in going 13 miles east or west the meridian has
changed 10'; and this is surely a sufficiently large correction
to make it worth while to apply it.
When running west, this correction is applied in the direc-
tion of the hands of a watch, and when running east, in the
opposite direction ; that is, having run west 13 miles by back
azimuth, then the pointing which appears north is really 10'
west of north, and the telescope must be shifted ic' around to
the right.
If the azimuth be corrected in this way, a survey can be
carried by back azimuth an indefinite distance, and still have
the entire survey referred to the true meridian.
419. The Angle of Convergence of Meiidlans is the
angle B in the equations given in the above formula. Then
tf = ;« sin Z,*
where n is the angular change in degrees of longitude, and L
is the latitude of the place.
For Z= 30°, sin L = i; or, in latitude 30*^ a change of
longitude of one degree changes the direction of the meridian
by 30 minutes.
For L = 40°, sin L = 0.643 ; or, a change of longitude of
one degree changes the direction of the meridian by 0.643 of
60 minutes, or 38.6 minutes, being approximately 40 minutes.
ForZ-= 50°, sin £=0.766; or, in going east or west one
^■^■— ■ — .
* From Eq. (G), p. 701, when cos i ^ Z is taken as unity.
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MAP-LETTERING AND TOPOGRAPHICAL SYMBOLS. 62<^
degree, the meridian changes 0.766 X 60 minutes = 46 min-
utes, or approximately 50 minutes.
Therefore we may have the approximate rule, thai a change
of longitude of one degree changes the azimuth by as many
minutes as equals the degrees of latitude of the place. This
rule gives results very near the truth between plus and minus
40° latitude, that is, over an equatorial belt 80 degrees in
width,
II. MAP-LETTERING AND TOPOGRAPHICAL SYMBOLS.
420. Map-Lettering. — The best-drawn map may have its
appearance ruined by the poor skill or bad taste displayed in
the lettering. The letters should be simple, neat, and dignified
in appearance, and have their size properly proportioned to the
subject. The map is lettered before the topographical symbols
are drawn. When a map is drawn for popular display, some
ornamentation may be allowed in the title ; but even then,
the lettering on the map itself should be plain and simple.
When the map is for official or professional use, even the title
should be made plain.
On Plate IV. are given several sets of alphabets which are
well adapted to map work. Of course the size should vary
according to the scale of the map and the subject, as shown on
Plate V. It is a good rule to make all words connected with
water in italics. The small letters in stump writing will be
found very useful, and these should be practised thoroughly.
The italic capitals go with these small letters also.
In place of the system of letters above described, and
which has heretofore been almost exclusively used for map-
ping purposes, a new system, called " round writing," may be
used. A text-book on this method, by F. Soennecken, is pub-
lished by Messrs. Kueffel & Esser, New York. This work is
done with blunt pens, all lines being made with a single stroke.
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630 BUkVEyiMC.
It looks well when well done, and requires but a small fraction
of the time required to make the ordinary letters. For work-
ing drawings and field maps it is especially adapted.
421. Topographical Symbols. — In topographical repre-
sentation, where elevations have been taken sufficiently num-
erous and accurate, the outline of the ground should be rep-
resented by contours rather than by hachures, or hill shading,
which simply gives an approximate notion of the slope of the
ground, but no indication of its actual elevation. Where the
ground has so steep a slope that the contour lines would fall
one upon another, it is well here to put in shading-lines, as
shown on Plate III. The water surfaces and streams may be
water-lined in blue, or left white. The contour lines over al-
luvial ground should be in brown (crimson and burnt sienna),
while those over rocky and barren ground should be in black.
This is a very simple and effective method of showing the
character of the soil.
The practices of the government surveys should be fol-
lowed in the matter of conventional surface representation,
such as meadow, swamp, woodland, prairie, cane-brake, etc.,
with all their varieties. Some of these are given in the United
States Coast Survey Report for 1879 and 1883, while Plate III.
shows most of those used on the Mississippi River Survey.
Those shown in Plate II. are adapted to higher latitudes, and
are those used in the field-practice surveys at Washington
University. This plate is an exact copy of one of the annual
maps made from actual surveys by the Sophomore class. On
these the contours are all in black, for the purpose of photo
lithographing.
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TOPOGRAPHICAL PRACTICE SURVEY
SWEET SPRINGS MO.
by the
v^C^HOUOBE CLASS
mth«
POLYTECHNIC SCHOOL
of
WASHINGTON UNIVERSITY
1886
■ Drwvn by
O.F.P«w-«pii
Seale:-400 feet^ one inch.
9 Surytry was triads by th» Tlnorruit and Stadia method hosnd an n :^yjt»m
zlation and Spirit Lbv^Is. Thg Datum Plana is token lOO fte( h%low
n. Lat. of SB.' BB'- ^' . PteVn of Ma§^netie NM€dU= a'-4o!
m
1
JS
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Convention
TOPOCRAFlti
Drnwii ;\nd
KHWARH \fm n
il Si^ns for
ICAi MAPS,
[; lOOOO .
lolo - lithographing.
PLATE ffl.
* V
>^1
' ' I *■.» PI II II 1 1
^ > ^Tt % * » ' > ,-'/■ '•-■ * ^ * . j r ^ — T |« =^l
s-
-l^A
^^ty\
fe.^i
--^_JiiL.
: %■•*■% WV'>''* '^j,^.,,^-^^ .__^__™,|
y-^^V^^ *^:V*iyi_4^*2 Mill '
ri^raved by
)H. T. E. Detroit
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8AMPLE PORTION OF THE CONTOUR MAPS PUBLISHED BY THE U. a GEOLOQIOAL SURVEY. I
SHOWINO THC AOVANTAQt Of OOLORIO OONTOUR AND STREAM LINIS.
PLATE
A PORTION OF THC OUMBCRLANO PLATCAU. IN W. VA.
OONTOUR INTCRVAL 100 FCCT.
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APPENDICES.
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APPENDIX A.
THE JUDICIAL FUNCTIONS OF SURVEYOIIS.
BY JUSTICE COOLEY OF THE MICHIGAN SUPREME COURT.
When a man has had a training in one of the exact sciences, where
every problem within its purview is supposed to be susceptible of accu-
rate solution, he is likely to be not a little impatient when he is told that,
under some circumstances, he must recognize inaccuracies, and govern
his action by facts which lead him away from the results which theoreti-
cally he ought to reach. Observation warrants us in saying that this re-
mark may frequently be made of surveyors.
In the State of Michigan all our lands are supposed to have been
surveyed once or more, and permanent moniiments fixed to determine
the boundaries of those who should become proprietors. The United
States, as original owner, caused them all to be surveyed once by sworn
officers, and as the plan of subdivision was simple, and was uniform over
a large extent of territory, there should have been, with due care, few or
no mistakes; and long rows of monuments should have been perfect
guides to the place of any one that chanced to be missing. The truth
unfortunately is that the lines were very carelessly run, the monuments
inaccurately placed ; and, as the recorded witnesses to these were many
times wanting in permanency, it is often the case that when the monument
was not correctly placed it is impossible to determine by the record, with
the aid of anything on the ground, where it was located. The incorrect
record of course becomes worse than useless when the witnesses it refers
to have disappeared.
It is, perhaps, generally supposed that our town plats were more ac-
curately surveyed, as indeed they should have been, for in general there
can have been no difficulty in making them sufficiently perfect for all
practical purposes. Many of them, however, were laid out in the woods;
some of them by proprietors themselves, without either chain or com-
pass, and some by imperfectly trained surveyors, who, when land was
cheap, did not appreciate the importance of having correct lines to deter-
mine boundaries when land should become dear. The fact probably is
that town surveys are quite as inaccurate as those made under authority
of the general government.
It is now upwards of fifty years since a major part of the public sur-
veys in what is now the State of Michigan were made under authority of
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634 SURVEYING.
the United States. Of the lands south of Lansing, it is now forty years
since the major part were sold and the work of improvement begun. A
generation has passed away since they were converted into cultivated
farms, and few if any of the original corner and quarter stakes now re-
main.
The corner atid quarter stakes wete ofteti nothing but green Sticks
driven into the ground. Stones might be put around or over these if
they were handyj but often they wete not, and the witness trees must be
relied upon after the stake was gone. Too often the first settlers \vt. e
careless in fixing their lines with accuracy while monuments remainedi
and an irregular brush fence, or something equally untrustworthy, niliy
have been rfelied upon to keep in mind where the blazed line once was.
A fire running through this might sweep it away, and if nothingwere sub-
stituted in its place, the adjoinmg proprietors might in a few years be
found disputing over their lines, and perhaps rushing into litigation, as
soon as they had occasion to cultivate the land along the boundary.
If now the disputing parties call in a surveyor, it is not likely that any
one summoned would doubt or question that his duty was to find, if
possible, the place of the original stakes which determined the boundary
line between the proprietors. However erroneous, may have been the
original survey, the monuments that were set miUt nevertheless govern,
even though the effect be to make one half-quarter section ninety acres
and the one adjoining but seventy; for parties buy or are supposed to
buy in reference to those monuments, and are entitled to what is within
their lines, and no more, be it more or less. Mclver v. Walker, \ Whea-
ton's Reports, 444; Laud Co. v. Saunders, 103 U. S. Reports, 316; Cot-
iingham v. Parr, 93 111. Reports. 233 ; Bunion v. Cardwell, 53 Texas Re»
ports, 408: Watson v. Jones, 85 Penn. Reports, 117.
While the witness trees remain there can generally be no difficulty in
determining the locality of the stakes. When the witness trees are
gone, so that there is no longer record evidence of the monuments, it is
remarkable how many there are who mistake altogether the duty that
now devolves upon the surveyor. It is by no means uncommon that we
find men whose theoretical education is supposed to make them experts
who think that when the monuments are gone, the only thing to be done
is to place new monuments where the old ones should have been, and
where they would have been if placed correctly. This is a serious mis-
take. The problem is now the same that it was before : to ascertain, by
the best lights of which the case admits, where the original lines were.
The mistake at)ove alluded to is supposed to have found expression in
our legislation ; though it is possible that the real intent of the act to
which we shall refer is not what is commonly supposed,
An act passed in 1869, Compiled Laws, § 593, amending the laws re^
specting the duties and powers of county surveyors, after providing for
the case of corners which can be identified by the original field-notes c r
other unquestionable testimon3^ directs as follows:*
" Second. Extinct interior section-corners must be re-established at.
the intersection of two right lines joining the nearest known points oxx
the original section lines east and west and north and south of it.
* For ibc U. 3. niles governing this subject sec Appendix I, page 736,
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APPENDIX A. 635
" Third. Any extinct quarter-section corner, except on fractional lines,
must be re-established equidistant and in aright line between the section
corners; in all other cases at its proportionate distance between the
nearest original corners on the same line."
The corners thus determined, the surveyors are required to perpetu-
ate by noting bearing trees when timber is near.
To estimate properly this legislation, we must start with the admit-
ted and unquestionable fact that each purchaser from government bought
snch land as. was within the original boundaries, and unquestionably
ovvned it up to the time when the monuments became extinct. If the
monument was set for an interior-section corner, but did not happen to
be "at the intersection of two right lines joining the nearest known
points on the original section lines east and west and north and sciith
of it," it nevertheless determined the extent of his possessions, and he
gained or lost according as the mistake did or did not favor him.
It will probably be admitted that no man loses title to his land oran>
part thereof merely because the evidences become lost or uncertain. It
may become more difficult for him to establish it as against an adverse
claimant, but theoretically the right remains; and it remains as a poten-
tial fact so long as he can present .better evidence than any other person.
And it may often happen that, notwithstanding the loss of all trace of a
section corner or quarter stake, there will still be evidence from which any
surveyor will be able to determine with almost absolute certainty where
the original boundary was between the government subdivisions.
There are two senses in which the word extinct may be used in this
connection : one the sense of physical disappearance ; the other the
sense of loss of all reliable evidence. If the statute speaks of extinct
corners in the former sense, it is plain that a serious mistake was made
in supposing that surveyors could be clothed with authority to establish
new corners by an arbitrary rule in such cases. As well might the stat-
ute declare that if a man lose his deed he shall lose his land altogether.
But if by extinct corner is meant one in respect to the actual location
of which all reliable evidence is lost, then the following remarks are per-
tinent:
1. There would undoubtedly be a presumption in such a case that
the corner was correctly fixed by the government surveyor where the
field notes indicated it to be.
2. Bnt this is only a presumption, and may be overcome by any satis-
factory evidence showing that in fact it was placed elsewhere.
3. No statute can confer upon a county surveyor the power to "estab-
lish " corners, and thereby bind the parties concerned. Nor is this a
question merely of conflict between State and Federal law ; it is a ques-
tion of property right. The original surveys must govern, and the laws
under which they were made must govern, because the land was bought
in reference to them ; and any legislation, whether State or Federal, that
should have the effect to change these, would be inoperative, because
disturbing vested rights.
4.. In any case of disputed lines, unless the parties concerned settle
the controversy by agreement, the determination of it is necessarily a
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636 ^Ul^lVEYiivG,
judicial act, and it must proceed upon evidence, and give full oppor-
tunity for a hearing. No arbitrary rules of survey or of evidence can
be laid down whereby it can be adjudged.
The general duty of a surveyor in such a case is plain enough. He
is not to assume that a monument is lost until after he has thoroughly
sifted the evidence and found him.self uraM^. to trace it. Even then he
should hesitate long before doing anything to the disturbance of settled
possessions. Occupation, especially if long continued, often affords very
satisfactory evidence of the original boundary when no other is attain-
able ; and the surveyor should inquire when it originated, how, and why
the lines were then located as they were, and whether a claim of title
has always accompanied the possession, and give all the facts due force
as evidence. Unfortunately, it is known that surveyors sometimes, in
supposed obedience to the State statute, disregard all evidences of occu-
pation and claim of title, and plunge whole neighborhoods into quarrels
and litigation by assuming to "establish " corners at points with which
the previous occupation cannot harmonize. It is often the case that
where one or more corners are found to be extinct, all parties concerned
have acquiesced in lines which were traced by the guidance of some
other corner or landmark, which may or may not have been trustworthy;
but to bring tliese lines into discredit when the people concerned do not
question them not only breeds trouble in the neighborhood, but it must
often subject the surveyor himself to annoyance and perhaps discredit,
since in a legal controversy the law as well as common-sense must declare
that a supposed boundarj^ line long acquiesced in is better evidence of
where the real line should be than any survey made after the original
monuments have disappeared. Stewart vs, Carleton, 31 Mich. Rep>orts.
270; Die hi vs. Zanger, 39 Mich. Reports, 601 ; Dupont vs. Starring, 42
Mich. Reports, 492. And county surveyors, no more than any others,
can conclude parties by their surveys.
The mischiefs of overlooking the facts of possession must often appear
in cities and villages. In towns the block and lot stakes soon disapp>ear ;
there are no witness trees and no monuments to govern except such as
have been put in their places, or where their places were supposed to be.
The streets are likely to be soon marked off by fences, and the lots in a
block will be measured off from these, without looking farther. Now it
may perhaps be known in a particular case that a certain monument still
remaining was the starting-point in the original survey of the town plat;
or a surveyor settling in the town may take some central point as the
point of departure in his surveys, and assuming the original plat to be
accurate, he will then undertake to find all streets and all lots by course
and distance according to the plat, measuring and estimating from his
point of departure. Tliis procedure might unsettle every line and every
monument existing by acquiescence in the town ; it would be very likely
to change the lines of streets, and raise controversies everywhere. Yet
this is what is sometimes done ; the surveyor himself being the first
p>erson to raise the disturbing questions.
Suppose, for example, a particular village street has been located by
acquiescence and use for many years, and the proprietors in a certain
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A^i-JiNDIX A, 637
biocK nave laid off their lots in reference to this practical location.
Two lot-owners quarrel, and one of them calls in a surveyor that he may
be sure that his neighbor shall not get an inch of land from him. This
surveyor undertakes to make his survey accurate, whether the original
was, or not, and the first result is, he notifies the lot-owners that there is
error in the street line, and that all fences should be moved, say, one foot
to the east. Perhaps he goes on to drive stakes through the block ac-
cording to this conclusion. Of course, if he is right in doing this, all
lines in the village will be unsettled ; but we will limit our attention to
the single block. It is not likely that the lot-owners generally will allow
the new survey to unsettle their possessions, but there is always a prob-
ability of finding some one disposed to do so. We shall then have a
lawsuit; and with what result?
It is a common error that lines do not become fixed by acquiescence
in a less time than twenty years. In fact, by statute, road lines may be-
come conclusively fixed in ten years; and there is no particular time
that shall be required to conclude private owners, where it appears that
ihey have accepted a particular line as their boundary, and all concerned
have cultivated and claimed up to it. McNamara vs. Seaton, 82 111. Re-
port<, 498; Bunce vs. Btdwell, 43 Mich. Reports, 542. Public policy re-
quires that such lines be not lightly disturbed, or disturbed at all after
the lapse of any considerable time. The litigant, therefore, who in such
a case pins his faith on the surveyor, is likely to suff^er for his reliance,
and the surveyor himself to be mortified by a result that seems to im-
peach his judgment.
Of course nothing in what has been said can require a surveyor to
conceal his own judgment, or to report the facts one way when he be-
lieves them to be another. He has no right to mislead, and he may
rightfully express his opinion that an original monument was at one
place, when at the same time he is satisfied that acquiescence has fixed
the rights of parties as if it were at another. But he would do mischief
if he were to attempt to "establish" monuments which he knew would
tend to disturb settled rights; the farthest he has a right to go, as an
officer of the law, is to express his opinion where the monument should
be, at the same time that he imparts the information to those who em-
ploy him, and who might otherwise be misled, that the same authority
that makes him an officer and entrusts him to make surveys, also allows
parties to settle their own boundary lines, and considers acquiescence in
a particular line or monument, for any considerable period, as strong, if
not conclusive, evidence of such settlement. The peace of the com-
munity absolutely requires this rule. Joyce vs. Williams, 26 Mich. Re-
ports, 332. It is not long since that, in one of the leading cities of the
State, an attempt was made to move houses two or three rods into a
street, on the ground that a survey under which the street had been
located for many years had been found on more recent survey to be
erroneous.
From the foregoing it will appear that the duty of the surveyor where
boundaries are in dispute must be varied by the circumstances, i. He
is to search for original monuments, or for the places where they w6re
39
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638 SURVEYING,
originally located, and allow these to control if he finds them, unless he
has reason to believe that agreements of the parties, express or implied,
have rendered them unimportant. By monuments in the case of gov-
ernment surveys we mean of course the corner and quarter stakes:
blazed lines or marked trees on the lines are not monuments ; they are
merely guides or finger-posts, if we may use tlie expression, to inform us
with more or less accuracy where the monuments may be found. 2. If
the original monuments are no longer discoverable, the question of loca-
tion becomes one of evidence merely. It is merely idle for any State
statute to direct a surveyor to locate or "establish" a corner, as the place
of the original monument, according to some inflexible rule. The sur-
veyor on the other hand must inquire into all the facts ; giving due prom-
inence to the acts of parties concerned, and always keeping in mind,
first, that neither his opinion nor his survey can be conclusive upon
parties concerned ; second, that courts and juries may be required to fol-
low after the surveyor over the same ground, and that it is exceedingly
desirable that he govern his action by the same ligiits and rules that will
govern theirs. On town plats if a surplus or deficiency appears in a
block, when the actual boundaries are compared with the original figures,
and there is no evidence to fix the exact location of the stakes which
marked the division into lots, the rule of common-sense and of law is
that the surplus or deficiency is to be apportioned between the lots, on
an assumption that the error extended alike to all parts of the block.
O'Brien vs. McGraney 29 Wis. Reports, 446; Quinnin vs. Reixers, 46
Mich. Reports, 605.
It is always possible when corners are extinct that the surveyor may
usefully act as a mediator between parties, and assist in preventing legal
controversies by settling doubtful lines. Unless he is made for this pur-
pose an arbitrator by legal submission, the parties, of course, even if they
consent to follow his judgment, cannot, on the basis of mere consent, be
compelled to do so; but if he brings about an agreement, and they carry
it into eflfect by actually conforming their occupation to his lines, the
action will conclude them. Of course it is desirable that all such agree-
ments be reduced to writing; but this is not absolutely indispensable if
they are carried into effect without.
Meander IJnes.—ThQ subject to which allusion will now be made is
taken up with some reluctance, because it is believed the general rules
are familiar. Nevertheless it is often found that surveyors misapprehend
them, or err in their application: and as other interesting topics are
somewhat connected with this, a little time devoted to it will probably
not be altogether lost. The subject is that of meander lines. These
are lines traced along the shores of lakes, ponds, and considerable rivers
as the measures of quantity when sections are made fractional by such
waters. These have determined the price to be paid when government
lands were bought, and perhaps the impression still lingers in some
minds that the meander lines are boundary lines, and all in front of
them remains unsold. Of course this is erroneous. There was never
any doubt that, except on the largfe navigable rivers, the boundary of the
owners of the banks is the middle line of the river; and while some
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APPENDIX A, 639
courts have held that this was the rule on all fresh-water streams, large
and small, others have held to the doctrine that the title to the bed of the
stream below low-water mark is in the State, while conceding to the
owners of the banks all riparian rights. The practical difference is not
very important. In this State the rule that the centre line is the bound-
ary line is applied to all our great rivers, including the Detroit, varied
somewhat by the circumstance of there being a distinct channel for
navigation in some cases with the stream in the main shallow, and also
sometimes by the existence of islands.
The troublesome questions for surveyors present themselves when the
boundary line between two contiguous estates is to be continued from the
meander line to the centre line of the river. Of course the original sur-
vey supposes that each purchaser of land on the stream has a water-front
of llie length shown by the field-notes ; and it is presumable that he
bought this particular land because of that fact. In many cases it now
happens that the meander line is left some distance from the shore by
the gradual change of course of the stream or diminution of the flow
of water. Now the dividing line between two government subdivisions
might strike the meander line at right angles, or obliquely ; and in some
cases, if it were continued in the same direction to the centre line of the
river, might cut off from the water one of the subdivisions entirely, or at
least cut it off from any privilege of navigation, or other valuaole use
X)f the water, while the other might have a water-front much greater
than the length of a line crossing it at right angles to its side lines.
The effect might be that, of two government subdivisions of equal size
and cost, qpe would be of very great value as water-front property, and
the other comparatively valueless. A rule which would produce tfiis re-
sult would not be just, and it has not been recognized in the law.
Nevertheless it is not easy to determine what ought to be the correcc
rule for every case. If the river has a straight course, or one nearly so,
every man's equities will be preserved by this rule : Extend the line of
division between the two parcels from the meander line to the centre line
of the river, as nearly as possible at right angles to the general course of
the river at that point. This will preserve to each man the water front
which the field-notes indicated, except as changes in the water may have
affected it, and the only inconvenience will be that the division line be-
tween different subdivisions is likely to be more or less deflected where it
strikes the meander line.
This is the legal rule, and it is not limited to government surveys, but
applies as well to water lots which appear as such on town plats. Bay
City Gas Light Co. v. The Industrial Works, 28 Mich. Reports, 182. It
often happens, therefore, that the lines of city lots bounded on navigable
streams are deflected as they strike the bank, or the line where the bank
was when the town was first laid out.
When the stream is very crooked, and especially if there are short
bends, so that the foregoing rule is incapable of strict application, it is
sometimes very difficult to determine what shall be done ; and in many
cases the surveyor may be under the necessity of working out a rule for
himself. Of course his action cannot be conclusive; but if he adopts one
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640 SURVEYIXG.
that follows, as nearly as the circumstances ;vli; aamit, the general rule
above indicated, so as to divide as near as may be the bed of the stream
among the adjoining owners in proportion to their lines upon the shore,
his division, being that of an expert, made upon the ground and with all
available lights, is likely to be adopted as law for the case. Judicial de-
cisions, into which the surveyor would find it prudent to look under such
circumstances, will throw light upon his duties and may constitute a suf-
ficient guide when peculiar cases arise. Each riparian lot-owner ought to
have a line on the legal boundary, namely, the centre line of the stream,
proportioned to the length of his line on the shore ; and the problem in
each case is, how this is to be given him. Alluvion, when a nver imper-
ceptibly changes its course, will be apportioned by the same rules.
The existence of islands in a stream, when the middle line constitutes
a boundary, will not affect the apportionment unless the islands were
surveyed out as government subdivisions in the original admeasurement.
Wherever that was the case, the purchaser of the island divides the bed
of the stream on each side with the owner of ihe bank, and his rights
also extend above and below the solid ground, and are limited by the
peculiarities of the bed and the channel. If an island was not surveyed as
a government subdivision previous to the sale of the bank, it is of course
impossible to do this for the purposes of government sale afterwards, for
the reason that the rights of the bank owners are fixed by their purchase :
when making that, tiiey have a right to understand that alljand between
the meander lines, not separately surveyed and sold, will pass with the
shore in the government sale ; and having this right, anything which
their purchase would include under it cannot afterward be taken from
them. It is believed, however, that the federal courts would 4iot recog-
nize the applicability of this rule to large navigable rivers, such as those
uniting the great lakes.
On all the little lakes of the State which are mere expansions near
their mouths of the rivers passing through them — such as the Muskegon.
Pere Marquette and Manistee — the same rule of bed ownership has been
judicially applied that is applied to the rivers themselves ; and the divi-
sion lines are extended under the water in the same way. Rice v. Ruddi-
man, 10 Mich., 125. If such a lake were circular, the lines would con-
verge to the centre ; if oblong or irregular, there might be a line in the
middle on which they would terminate, whose course would bear some
relation to that of the shore. But it can seldom be important to follow
the division line very far under the water, since all private rights are sub-
ject to the public rights of navigation and other use, and any private use
of the lands inconsistent with these would be a nuisance, and punishable
as such. It is sometimes important, however, to run the lines out for
some considerable distance, in order to determine where one may law-
fully moor vessels or rafts, for the winter, or cut ice. The ire crop that
forms over a man's land of course belongs to him. Lorman v. Benson^ 8
Mich.. 18; People's Ice Co. v. Steamer Excelsior, recently decided.
What is said above will show how unfounded is the notion, which is
sometimes advanced, that a riparian proprietor on a meandered river may
lawfully raise the water in the stream without liability to the proprietors
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APPENDIX A. 641
above, provided he does not raise it so that it overflows the meander line.
The real fact is that the meander line has nothing to do with such a case,
and an action will He whenever he sets back the "water upon the proprie-
tor above, whether the overflow be below the meander lines or above
them.
As regards the lakes and ponds of the State, one may easily raise
questions that it would be impossible for him to settle. Let us suggest
a few questions, some of which are easily answered, and some not :
1. To whom belongs the land under these bodies of water, where they
are not mere expansions of a stream flowing through them ?
2. What public rights exist in them ?
3. If there are islands in them which were not surveyed out and sold
by the United States, can this be done now.^
Others will be suggested by the answers given to these.
It seems obvious that the rules of private ownership which are applied
to rivers cannot be applied to the great lakes. Perhaps it should be held
that the boundary is at low-water mark, but improvements beyond this
would only become unlawful when they became nuisances, Islands in
the great lakes would belong to the United States until sold, and might
be surveyed and measured for sale at any time. The right to take fish in
the lakes, or to cut ice, is public like the right of navigation, but is to be
exercised in such manner as not to interfere with the rights of shore
owners. But so far as these public rights can be the subject of ownership,
they belong to the State, not to the United States ; and, so it is believed,
does the bed of a lake also. Pollard v. Hagan, 3 Howard's U. S. Reports.
But such rights are not generally considered proper subjects of sale, but,
like the right to make use of the public highways, they are held by the
State in trust for all the people.
What is said of the large lakes may perhaps be said also of many of
the interior lakes of the State; such, for example, as Houghton, Higgins,
Cheboygan, Burt's, Mullet, Whitmore, and many others. But there are
many little lakes or ponds which are gradually disappearing, and the
shore proprietorship advances /rtr//fl55« as the waters recede. If these
are of any considerable size — say, even a mile across — there may be ques-
tions of conflicting rights which no adjudication hitherto made could
settle. Let any surveyor, for example, take the case of a pond of irregu-
lar form, occupying a mile square or more of territory, and undertake to
determine the rights of the shore proprietors to its bed when it shall
totally disappear, and he will find he is in the midst of problems such as
probably he has never grappled with, or reflected upon before. But the
general rules for the extension of shore lines, which have already been
laid down, should govern such cases, or at least should serve as guides in
their settlmeent. Note. — Since this address was delivered some of these
questions have received the attention of the Supreme Court of Michigan
in the cases of Richardson v. Prentiss, 48 Mich. Reports, 88, and Backus
V. Detroit, Albany Liiw Journal, vol. 26, p. 428.
Where a pond is so small as to be included within the lines of a pri-
vate purchase from the government, it is not believed the public have any
rights in it whatever. Where it is not so included, it is believed they have
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SURVEYING,
rights of fishery, rights to take ice and water, and rights of navigation for
business or pleasure. This is the common belief, and probably the Just
one. Shore rights must not be so exercised as to disturb these, and the
States may pass all proper laws for their protection. It would be easy
with suitable legislation to preserve these little bodies of water as perma-
nent places of resort for the pleasure and recreation of the people, and
there ought to be such legislation.
If the State should be recognized as owner of the beds of these small
lakes and ponds, it would not be owner for the purpose of selling. It
would be owner only as a trustee for the public use; and a sale would be
inconsistent with the right of the bank owners to make use of the water
in its natural condition in connection with their estates. Some of them
might be made salable lands by draining ; but the State could not drain,
even for this purpose, against the will of the shore owners, unless their
rights were appropriated and paid for.
Upon many questions that might arise between the State as owner of
the bed of a little lake and the shore owners, it would be presumptuous to
express an opinion now, and fortunately the occasion does not require it.
I have thus indicated a few of the questions with which surveyors may
now and then have occasion to deal, and to which they should bring good
sense and sound judgment. Surveyors are not and cannot be judicial
oflicers, but in a great many cases they act in a quasi judicial capacity
with the acquiescence of parties concerned ; and it is important for them
to know by what rules ihey are 10 be guided in the discharge of their
judicial functions. What I have said cannot contribute much to their
enlightenment, but I trust will not be wholly without value.
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APPENDIX B.
MANUAL OF INSTRUCTIONS FOR THE SURVEY OF THE
MINERAL LANDS OF THE UNITED STATES.*
GENERAL PROVISIONS.
1. Under section 2334, U. S. Rev. Stats, (see Appendix B hereoO, the
United States Surveyor-General " may appoint in each land district con-
taining mineral landt as many competent surveyors as shall apply for
appointment to survey mining claims."
2. Capable parties desiring such appointments should therefore file
their applications with the Surveyor-General for the district wherein ap-
pointment is asked, who will furnish all information necessary.
3. Deputies may at the same time hold commissions in more than
one State or land district. (20 L. D., 163.)
4. All appointments of deputy mineral surveyors must be submitted
to the Commissioner of the General Land Office for approval.
5. The Surveyors-General have authority to suspend or revoke the
commissions of their deputy mineral surveyors for cause. Before final
action, however, the matter should be submitted to the Commissioner of
the General Land Office for approval.
Deputies will be allowed the right of appeal from the action of the
Surveyor-General in the usual manner. Such appeal should be filed with
the Surveyor-General, who will at once transmit the same, with a full
report, to the General Land Office. (20 L. D., 2S3.)
6. Neither the Surveyor-General nor the Commissioner of the General
Land Office has jurisdiction to settle differences, relative to the payment
of charges for fieldwork, between deputy mineral surveyors and claimants.
These are matters of private contract and must be enforced in the ordinary
manner, i.e., in the local courts.
The Department has, however, authority to investigate charges affect-
ing the official actions of deputy mineral surveyors, and will, on sufficient
cause shown, suspend or revoke absolutely the commission of the deputy.
7. The Surveyors-General should appoint '" as many " competent
deputy mineral surveyors as apply for appQintment. in order that ch'mants
may have a choice of deputies, and be enabled to have their work done
on the most advantageous terms.
8. The schedule of charges for office work should be as low as is pos-
sible. No additional charges should be made for orders for amended sur-
veys, unices the necess'ty therefor is clearly the fnult of the claimant, or
considerable a'^dit'onal office work results therefrom.
* Reprinted in full from the official publication issued in 1895 by the Commi>^sioner of the
GeneralLand Office, Washington, D.C.
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644 SURVEYING.
In cases where the error in the ori'^inal s'rvey is due to the careless-
ness or neglect of the deputy mineral surveyor who made it, he should
be required to make the necessary corrections in the field at his own ex-
pense, and the Surveyor-General should advise him that the penalty fbr
failure to comply with instructions within a specified time will be the
suspension or revocation of his commission.
9. These instructions are subject to the limitations of section 2324,
U. S. Rev. Stats., so far as the same refers to local laws and customs.
INSTRUCTIONS FOR SURVEYS.
1. All official communications must be addressed to the Surveyor-
General.* You will always refer to the date and subject-matter of the
letter to which you reply, and when a mineral claim is the subject of corre-
spondence, you will give the name and survey number.
2. You should keep a complete record of each survey made by you,
and the facts coming to your knowledge at the time, as well as copies of
all your field notes, reports, and oiTicial correspondence, in order that
such evidence may be readily produced when called for at any future
time.
3. Field notes and other reports must be written in a clear and legible
hand or typewritten, in noncopying ink, and upon the proper blanks fur-
nished you gratuitously by the Surveyor-General's Office upon applica-
tion therefor. No interlineations or erasures will be allowed; and no abbre-
viations or symbols must be used, except such as are indicated in the
specimen field notes.
4. No return by you will be recognized as official, unless it is over your
signature as a U. S. deputy mineral sur.eyor, and made in pursuance of
a special order from the Surveyor-General's OlTice. After you have re-
ceived an order for survey, you are required to make the survey, and re-
turn correct field no es thereof to the Surveyor-General's Office without
delay.
5. The claimant is required, in all cases, to make satisfactory arrange-
ments with you for the payment for your services and those of your assist-
ants in making the survey, as the United States will not be held respon-
sible for the same. You will call the attention of applicants for mineral-
survey orders to the requirements of paragraph 14 of the circular, Appen-
dix A. (Sec. 2334, U. S. Rev. Stats.: Par. 98, Mining Circular, Decem-
ber 10, 1891 — see Appendix B hereof.)
6. You will promptly notify the Survevor-General's Office of any
change in your post-ofiice address. (20 L. D., 163.)
NOT TO ACT AS ATTORNEY.
7. You are precluded from acting, either directly or indirectly, as
attorney in mineral claims. Your duty in any particular case ceases when
you have executed the survey and returned the field notes and preliminary
plat, with your report, to the Surveyor-General. You will not be allowed
to prepare for the mining claimant the papers in support of his applica-
tion for patent, or otherwise perform the duties of an attorney before
the land ofiice in connection w.4h a mining claim. You are not permitted
to combine the duties of surveyor and notary public in the same case by
♦ For list of Surveyors-General io roiuing dislricis, see pa^c 684.
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APPENDIX B. 645
aditi mistering oaths to the parties in interest, but as a notary public you
may administer the oaths to your assistants in making the survey. Other-
wise you must have absolutely nothing to do with the case, except in your
official capacity as surveyor. You will make no survey of a mineral xlaim
in which you hold an inlerest, nor will you employ chainmen interested
therein in any manner. (Par. loi, Mining Circular, December 10, 1891 —
see Appendix B hereof; 13 C. L. O., 608.)
THE FIELDWORK.
8. The survey made and reported must, in every case, be an actual
survey on the ground in full detail, made by you in person after the re-
ceipt of the order, and without reference to any knowledge you may have
previously acquired by reason of having made the location survey or
otherwise, and must show the actual facts existing at the time. This pre-
cludes you from calculating the connections to corners of the public sur-
vey and location monuments, or any other lines of your survey through
prior surveys made by others, unless it is satisfactorily shown in your re-
port that you have retraced such lines and found them to be correct. (6
L. D., 718; 7 L. D., 81.)
The term sun'cy in these instructions applies not only to the usual
fieldwork, but also to the examinations required for the preparation of
your affidavits of five hundred dollars expenditure, descriptive reports on
placer claims, and all other reports.
SURVEY AND LOCATION.
9. The survey of a mining claim may consist of several contiguous lo-
cations, but such survey must, in conformity with statutory requirements,
distinguish the several locations, and exhibit the boundaries of each. (5
L. D., 199; 6 L. D., 808.)
ID. The survey must be made in strict conformity with, or be embraced
within, the lines of the location upon which the order is based. If the
survey and location are identical, that fact must be clearly and distinctly
stated in your field no*cs. If not identical, a bearing and distance must be
given from each established corner of the survey to the corresponding
corner of the location, and the location corner must be fully described, so
that it can be identified. The lines of the location, as found upon the
ground, must be laid down upon the preliminary plat in such a manner
as to contrast and show their relation to the lines of survey. (I L. D., 581.)
II. In accordance with the principle that courses and distances must
give way when in conflict with fixed objects and monuments, you will not,
under any circumstances, change the corners of the location for the pur-
pose of making them conform to the description in the record. If the
difl^erence from the location be slight, it may be explained in the field
notes.
" The act of Congress of May 10, 1872, expressly provides that ' the
location must be distinctly marked upon the rround s ) that its bound-
aries can be readily traced,' and * that all records of mining claims here-
after made shall contain the naire cr names of the locators, the date of
the location, and such a description of the claim or claims located, by
reference to some natural object or permanent monument, as will identify
the claim/" (Sec 3324, U. S. R. S.; see Appendix B herewith.)
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646 SURVEYING,
" These provisions of the law must be strictly complied with in each
case to entitle the claimant to a survey and patent, and, therefore, should
a claimant under a location made subsequent to the passage of the act of
May ID, 1872, who has not complied with said requirements in regard to mark-
ing the location upon the ground and recording the sanw, apply for a survey, you
will decline to make it." (i L. D., 581.)
You will then report the facts to this office and await further instruc-
tions.
Should the survey be applied for under a location made prior to May
10, 1872, or under section 2332, U. S. Rev. Stats., in making the survey
thereof you will be governed by the special instructions accompanying the
order for survey.
No mining claim located subsequent to May 10, 1872, should exceed
the statutory limit in width on each side of the centre of vein or 1,500
feet in length, and all surveys must close within 50-100 feet in one
thousand feet, and the error must not be such as to make the location ex-
ceed the statutory limit, and in absence of other proof the discovery point
is held to be the centre of the vein on the surface. The course and length
of the vein should be marked upon the plat.
INSTRUMENT.
12. All mineral surveys must be made with a transit, provided with
a solar attachment, by which the meridian can be determined independ-
ently of the magnetic needle, and all courses must be referred to the true
meridian. The variation should be noted at each corner of the survey.
THE TRUE MERIDIAN.
12b. The true course of at least one line of each survey must be ascer-
tained by astronomical observations made at the time of the survey; the
data for determining the same and details as to how these data were ar-
rived at must be given. Or, in lieu of the foregoing, the survey must be
connected with some line the true course of which has been previously
established beyond question, and in a similar manner, by yourself, and,
when such lines exist, it is desirable in all cases that they should be used
as a proof of the accuracy of subsequent work. (For methods for deter-
mining the true meridian see pages 84 to 119, inclusive, J3eneral Survey-
ing Manual, 1894.)
CONNECTIONS.
13. Connect corner No. i of each location embraced in your survey
by course and distance with nearest corner of the public survey or with a
United States location monument, if the claim lies within two miles of
such corner or monument. If both are within the required distance, you
must connect with the nearest corner of the public survey. (7 L. D., 475;
Par. 45. Circular December 10, 1891 — Appendix B herewith.)
(a) You will make surveys and connections of mineral claims in sus-
pended townships so long as they remain suspended, in the same manner
as though the claims were upon unsurvcyed land, excert as hereinafter
specified, by connecting them with independent mineral mcnuTcnts. At
the same time, you will note the position of any public-land corner which
may be found in the neighborhood of the claim, so that, in case of the re-
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APPENDIX B. 647
lease of the township plat from suspension, the position of the claim can
be shown on the plat.
(6) A mineral survey must not be returned with its connection made
only with a corner of the public survey, where the survey of the town-
ship within which it is situated is under suspension, nor connected with a
mineral monument alone, when situated within the limits of a township,
the regularity and correctness of the survey of which is unquestioned.
(f) In making an official survey hereafter you will establish corner
No. I of each location embraced in your survey at the end nearest the
corner of the public survey or locating monument, unless good cause is
shown for its being placed otherwise. If connections are given to both a
corner of the public survey and location monument, corners Nos. i should
be placed at the end nearest the corner of the public survey.
14. When a boundary line of a claim intersects a section line, give
courses and distances from point of intersection to the Government
corners at each end of the half mile of section line so intersected.
^ LOCATION MONUMENTS.
15. In case your survey is situated in a district where there are no
corners of the public survey and no monuments within the prescribed
limits, you will proceed to establish a mineral monument, in the location
of which you will exercise the greatest care to insure permanency as to
site and construction.
The site, when practicable, should be some prominent point, visible
for a long distance from every direction, and should be so chosen that
the permanency of the monument will not be endangered by snow, rock, or
land slides, or other natural causes.
16. The location monument should consist of a stone not less than
thirty inches long, twenty inches wide, and six inches thick, set halfway
in the ground, with a conical mound of stone four feet high and six feet
base alongside. The letters U. S. L. M., followed by the consecutive
number of the monument in the district, must be plainly chiselled upon the
stone. If impracticable to obtain a stone of required dimensions, then a
post eight feet long, six inches square, set three feet in the ground, scribed
as for a stone monument, protected by a well-built conical mound of stone
of not less than three feet high and six feet base around it, may be used.
The exact point for connection must be indicated on the monument by a
X chiselled thereon; if a post is used, then a tack must be driven into the
post to indicate the point.
17. From the monument, connections by course and distance must be
taken to two or three bearing trees or rocks, and to any well-known and
permanent objects in the vicinity, such as the confluence of streams,
prominent rocks, buildings, shafts, or mouths of adits. Bearing trees
must be properly scribed B.T., and bearing rocks chiselled B. R., together
with the number of the location monument; the exact point on the tree
or stcne to which the connection is taken should be indicated by a cross
or other unmistakable mark. Bearings should also be taken to promi-
nent mountain peaks, and the approximate distance and directon ascer-
tained from the nearest town or mining camp. A detailed description of
the locating monument, with a topographical map of its location, should
be furnished this office.
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648 SURVEYING.
CORNERS.
18. Corners may consist of —
First — A stone at least 24 inches long set 12 inches in the ground,
with a conical mound of stone 1V3 feet high, 2 feet base, alongside, and
fetate kind of stone ?et for corner. A stone should always be used for a
corner when possible.
Second — A post at least 3 feet long by 4 inches square, set 18 inches in
the ground and surrounded by a substantial mound of stone or earth.
Third — A rock in pla:e.
19. .^11 corners must be established in a permanent and workmanlike
manner, and the corner and survey number must be neatly chiselled or
scribed on the sides facing tlie claim. The exact corner point must be
permanently indicated on the corner. When ai rock in place is used its di-
mensions above ground must be stated, and a cross chiselled at the exact
corner point.
20. In case the point for the corner be inaccessible or unsuitable, you
will establish a witness corner, which must be marked with the letters
W. C. in addition to the corner and survey number. The witness coMer
should be located upon a line of the survey and as near as possible to the
true corner, with which it must be connected by course and distance. The
reason why it is impossible or impracticable to establish a true corner
must always be stated in the field notes, and in running your next course
state whether you start from the true place for corner or 'from witness
corner.
21 The identity of all corners should be perpetuated by taking courses
and distances to bearing trees, rocks, and other objects, as prescribed in
the establishment of location monuments, and when no bearings are
given, state "no bearings available." Permanent objects should be taken
whenever possible.
22. If an official survey has been made within a reasonable distance in
the vicinity, you will run a connecting line to some corner of the same,
and connect in like manner with all conflicting surveys and claims, and
describe corner with which connection is made.
23. In survey of contiguous locations which are part of a consolidated
claim, where corners are common, mention bearings but once, but where
a corner is common to different claims, the required number of bearings
will be taken from each claim.
TOPOGRAPHY.
24. Note carefully ah topographical features of the claim, taking dis-
tances on your lines to intersections with all streams, gulches, ditches,
ravines, mountain ridges, roads, trails, etc., with their widths, courses,
and other data that may be required to map them correctly. If the claim
lies within a town site, locate all municipal improvements, such as V^-ocks,
streets, and buildings.
CONFLICTS.
25. If, in running the exterior boundaries of a claim, you find that two
surveys conflict, you will determine the courses and distances from the
established corners thereof, situate within the boundaries of your survey,
at which the exterior boundaries intersect eich other, and run all lines
necessary for the determination of the areas in conflict, both with sur-
veyed and unsurveycd cliims. Von will not, however, show conflicts with
unsurveyed claims unless the same are to be excluded.
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APPENDIX B.
649
26. Your attention is directed to paragraphs 50 and 51 of General
Land Office circular, dated December 10, 1891, as amended by circular of
November 7, 1895:
" 50. The rights granted to locators under section 2322, Revised Statutes, are restricted to
such locations on veins, lodes, or ledges as may t)e * situated on the public domain.'' In ap-
plications for lode claims where the survey conflicts with a prior valid lode claim, and the
ground in conflict is excluded, the applicant not only has no right to the excluded ground, but
he has no right to that portion of any vein or lode, the top or apex of which lies within such
excluded ground, unless his location was prior to May jo, 1872. His right to the lode claimed
terminates where the lode, in its onward course or strike, intersects the exterior boundary of
such excluded ground and passes within it. The end line of his survey should not, therefore,
be established beyond such intersection.
**5i. Where, however, tilfc lode claim for which survey; is being made, was located prior to
the conflicting claim, and such conflict is to be excluded, in order to include all ground not so
excluded the end line of the survey may be established within the conflicting lode claim, but
the line must be so run as not to extend any further into such conflicting claim than may be
necessary to make such end line parallel to the other end line, and at the same time embrace
the ground so held and claimed. The useless practice in such cases of extending ^<rM the side
lines of a survey into the conflicting claim, and establishing an end line wholly within it,
beyond a point necessary under the rule just stated, will be discontinued/^
EXPLANATION.
Location " A *' in the diagram represents a location for which survey
is applied for. As a location it conflicts with location ** B,** and the
claimant of " A " lays no claim to the conflicting area. In accordance
with the definition of locators' rights un-
der Sec. 2322, U. S. R. S., as given in
paragraph 50 of said circular of Decem-
ber 10, 1891, if location " B " was a valid
and subsisting location at the time location
'* A " was made, the locator of " A '* has
no right to extend the survey of his claim
beyond the point where his lode in its
onward course intersects the location
" B," and in such case the end lines of
the survey should be run through such
point of intersection, as represented by
lines ** a " '* b " on diagram.
If it is more desirable, however, the
south end line may be the side line of *' B "
within the §ide lines of '* A " as represented
by ** c " " y/' with which the north end
line must be made parallel.
This circular applies also to contiguous
locations belonging to one owner and sur-
veyed as a single claim.
If, on the contrary, when locafion " A "
was made the location *' B " was not
existent, and there was no other conflict,
the abandonment of the conflict with the
latter location, " B,** brings the case within
paragraph 51 of said circular.
To illustrate: " A " is the first and valid
location, afterward the ** B " location is
made, and for reasons satisfactory to
" A " the ground in conflict is relinquished to " B," yet the claimant of
" A " claims all the land outside of the conflict included in his location.
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650 SURVEYING.
The end line contemplated by the circular to include all ground claimed
as aforesaid, should be lun as represented by the line '* c ' *' d," the rest
of the lines of conflict being eliminated entirely. The survey must in this
case stop at the line ** c " " d.''
27. It will be particularly observed by you that the above provisions
of General Land Office circular dated December 10, 1891, are just as appli-
cable in the case of conflicting and overlapping locations embraced in
one survey as though the several locations were embraced in separate and
distinct surveys.
28. A lode claim that is divided into two parts by an intersecting pat-
ented mill site, placer, or agricultural entry, must be confined to that part
which contains the discovery shaft and improvements. (13 L. D., 146.)
29. The exterior lines of placer claims or niiil sites cannot be extended
over other claims and the conflicting areas excluded, as with lode claims,
it being the surface ground only, with side lines taken perpendicularly
downward, for which application is made. The survey must accurately
define the boundaries of the claim.
30. If, by reason of intervening surveys or claims, a placer survey
should be divided into separate tracts, you will preserve a separate series
of numbers for the corners of each location, and a consecutive series of
numbers for the corners of the tracts embraced in each; distinguishing
the detached portions as Tract A, Tract B, etc., connecting by course and
distance a corner of each tract with some corner of one previously de-
scribed. The provisions of this paragraph will also apply to the surveys
of mill sites.
LODE AND MILL SITE.
31. A lode and mill site claim in one survey will be distinguished by
the letters A and B following the number of the survey. The corners
of the mill site will be numbered independently of those of the lode. Cor-
ner No. I of the mill site must be connected with a corner of the lode
claim as well as with a corner of the public survey or United States loca-
tion monument.
FIELD NOTES.
32. In order that the results of your survey may be reported in a uni-
form manner, you will prepare your field notes and preliminary plat in
strict conformity with the specimen field notes and plats, which are made
part of these instructions. They are designed to furnish you all the needed
information concerning the manner of describing the boundaries, corners,
connection^, intersections, conflicts; and improvements, and stating the
variation, area, location, and other data connected with the survey of
mineral claims, and contain forms of affidavits for the deputy surveyor and
his assistants.
ZZ' When a placer claim includes lodes, or when several contiguous
placer or lode locations are included as one claim in one survey, you will
give to the corners of each location constituting the same a separate con-
secutive numerical designation, beginning with Corner No. i in each case.
In the former case you will first describe the placer claim in your field
notes.
34. Throughout the description of the survey, after each reference to
the lines or corners of a location, give the name thereof, and if unsurveyed
state the fact. If reference is made to a location included in a prior official
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APPENDIX R 651
survey, the survey number must be given, followed by the name of the
location. Describe your corners once only.
35. The total area of each location in a claim embraced by its exterior
boundaries, and also the area in conflict with each intersecting survey or
claim, should be stated; also the total area claimed. But when locations
of the survey conflict with each other, such conflicts should only be stated
in connection with the location from which the conflicting area is ex-
cluded.
36. You will state particularly whether the claim is upon surveyed or
unsurveyed public lands, giving in the former case the quarter section,
township, and range in which it is located, and in the latter, the township,
as near as can be determined. When upon surveyed lands the section lines
should be indicated by full lines and the quarter-section lines by dotted
lines.
Zl' The title page must contain the post-office address of the claimant
or his authorized agent.
EXPENDITURE OF FIVE HUNDRED DOLLARS.
38. In making out your certificate of the value oi the improvements,
you will follow the form prescribed in the specimen field notes.
39. Only actual expenditures and mining improvements made by the
claimant or his grantors, having a direct relation to the development of
the claim, can be included in your estimate. *' Labor or improvements
within the meaning of the statute are deemed to have been had on a
mining claim, whether it consists of one location or several, when the
labor is performed or the improvements are made for its development,
that is, to facilitate the extraction of the metals it may contain." (6 L. D.,
222.)
40. The expenditures required may be made from the surface or in
running a tunnel, drifts, or cross-cuts, for the development of the claim.
Improvements of any other character, such as buildings, machinery, or
roadways, must be excluded from your estimate unless you show clearly
that they are associated with actual excavations, such as cuts, tunnels,
shafts, etc., and are essential to the practical development of the surveyed
claim.
41. You will locate all mining and other improvements upon the claim
by courses and distances from corners of the survey, or from points on
the centre or side lines, specifying with particularity and detail the dimen-
sions and character of each, and the improvements upon each location
should be numbered consecutively, the point of discovery being always
No. I.
42. You will give in detail the value of each mining improvement in-
cluded in your estimate of expenditure, and when a tunnel or other im-
provement has been made for the development of other claims in con-
nection with the one for which survey is made, you must give the name,
ownership, and survey number, if any. of each claim to which a portion
or interest is credited, and the value of the portion or interest credited to
the claim. The value of improvements made upon other locations, or by
a former locator who has abandoned his claim, cannot be included in
your estimate, but should be described and located in your notes and plat.
43. In case of a lode and mill-site claim in the same survey, an expendi-
ture of five hundred dollars must be shown upon the lode claim only.
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652 SURVEYING,
44. When a survey embraces several locations held in common con-
stituting one entire claim, whether lode or placer, an expenditure of five
hundred dollars upon such entire claim embraced in the survey will be
sufficient, but in preparing your estimate of five hundred dollars expendi-
ture thereon you will observe the requirements of the decision of the
Commissioner of the General Land Office, dated June 11, 1890, quoted be-
low for your information :
** When two or more lode locations are embraced in one entry, and
the improvements on each lode are not of the value of $500, it must be
shown that a sum equal in value, of labor or improvements, has been ex-
pended for the common benefit of all those of which the improvements
do not equal that sum, with a satisfactory explanation of how and in what
manner such improvements tend to a common benefit." (See also 20
L. D.. 394.)
The explanatory statement in such cases should be given in your field
notes, or affidavit, at the conclusion of the description of the improve-
ments included in the estimate of expediture, and should be as full and
explicit as the facts in the case warrant, dealing only with the improve-
ments, conditions, and circumstances, as they actually existed at the time
of making the survey or examination, without reference to what is possi-
ble or what the claimants may intend to do.
45. Following your certificate you will locate and describe all other
improvements made by the claimant within the boundaries of the survey.
Those made by other parties, if any, whose names should be mentioned,
will be given in a separate description, following those of the claimants.
46. If the valve cf the labor and improvements upon a mineral claim
is less than five hundred dollars at the time of survey, you are author-
ized to file yonr affidavit of five hundred dollars expenditure at any time.
If the affidavit is made subsequent to the period of publication, it should
be shown, if practicable, when the improvements were made. The infor-
mation on which to base this affidavit must be derived by the deputy who
makes the actual survey from a careful examination upon the premises.
prelimiinvvry plat.
47. You will file with your field notes a preliminary plat on blank sent
you for that purpose, protracted on a scale of two hundred feet to an
inch, if practicable, in conformity with the specimen plat herewith. In
preparing plats make the top north. Copy of your calculations of areas
by double meridian distances, and of all triangulations or traverse lines,
must also be furnished. The lines of the claim surveyed, on this plat Snd
on all plats of approved surveys, should be heavier and show a contrast
with conflicting claims.
ERRORS.
48. You will also mention in your notes the discovery of any material
errors in prior official surveys, stating explicitly what lines are found to
be in error, and giving in express terms the courses and lengths thereof
as determined by you.
49. Whenever a survey has been reported in error, the deputy surveyor
who made it will be required to promptly make a thorough examination,,
upon the premises, and report the resuh, under oath, to the Surveyor-
General's office. In case he finds his survey in error, he will report in
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APPENDIX B. 653
detail all discrepancies with the original survey, and submit any explana-
tion he may have to offer as to the cause. If, on the contrary, he should
report his survey correct, a joint survey will be ordered to settle the dif-
ferences with the surveyor who reported the error.*
JOINT SURVEY.
50. A joint survey must be made within ten days after the date of
order, unless satisfactory reasons are submitted, under oath, for a post-
ponement.
51. The field work must in every sense of the term be a joint and not
a separate survey, and the observations and measurements taken with the
same instrument and chain, previously tested and agreed upon.
52. The deputy surveyor found in error or. if both are in error, the
one who reported the same will make out the field notes of the joint sur-
vey, which, after being duly signed and sworn to by both parties, must
be transmitted to the Surveyor-General's offtce. The surveyor found in
error will be required to pay all expenses of the joint survey and ten
dollars per day to the surveyor whose work is found to be substantially
correct, and either deputy shall have the right to require the other to
deposit the estimated amount of this expense with the Surveyor-General
before the joint work is begun.
AMENDED SURVEYS.
53. Inasmuch as amended surveys are ordered only by special in-
structions from the General Land Oifice, and the conditions and circum-
stances peculiar to each separate case, and the object sought by the re-
quired amendment, alone govern all special matters relative to the man-
ner of making such survey and the form and subject-matter to be em-
braced in the field notes thereof, but few general rules applicable to all
cases can be laid down.
54. The amended survey must be made in strict conformity with, or
be embraced within, the lines of the original survey. If the amended and
original surveys are identical, that fact must be clearly and distinctly
stated in your field notes. If not identical, a bearing and distance must
•The followiofir circular recently issued (iS^) by ihc Hon. Commissioner of the General
Land Office has somewhat changed the practice of reporting? out surveys in error :
*' When a mining claim has been surveyed and patented in accordance therewith, the land
described therein is disposed of, and so long- as the patent is outstanding the jurisdiction of
the Department in regard to that particular tract is terminated. It therefore follows that
land thus patented cannot be properly included in a subsequent patent, merely because years
afterwards a deputy mineral surveyor in making a subsequent survey repyorts to have found
the true corners of the old survey to occupy a different position from that reported in the
survey which was the basis for patent of the old claim. And the same thing is true as to
reported discrepancies as to the length and courses of lines of prior approved surveys.
'* Where such a state of things actually exists the owner of the new claim apolied for, who
desires to include an area in his claim, conveyed in a patent of an older claim, which as a
matter of fact is not embraced in the lines of the old claim as staked upon the ground, should
procure the surrender of the old patent by the proper method, through the courts if neces-
sary, and then show in a new patent of the old claim its true position as staked and thus
eliminate from the patent the areas desired not in conflict."
As a result of the above circular the deputy when he finds a prior approved survey in
error cannot report it out, but must give a tie to the position of the corners as they appear
by the records of the Surveyor-General's Office and not as they actually are on the ground.
At present the practice is to calculate ties to prior official surveys through the section corner
connections.
This ruling seems to a certain extent to be a violation of the old-established priDciple that
'* monuments hold over descriptions.*'
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6S4
SURVEYING,
be given from each established corner of the amended survey to the cor-
responding corner of the original survey. The lines of the original sur-
vey, as found upon the ground, must be laid down upon the preliminary
plat in such manner as to contrast and show their relation to the lines of
the amended survey.
55. The field notes of the amended survey must be prepared on the
same size and form of blanks as are the field notes of the original survey,
and the word *' amended " must be used before the word ** survey "
wherever it occurs in the field notes,
DESCRIPTIVE REPORTS ON PLACER CLAIMS.
56. By General Land Office circular approved December 10, 1891, par.
63 (see Appendix B hereof), you are required to make a full examination
of all placer claims at the time of survey, and file with your field notes a
descriptive report, in which you will describe:
(a) The quality and composition of the soil, and the kind and amount
of timber, and other vegetation.
{h) The locus and size of streams, and such other matter as may appear
upon the surface of the claims.
(f) The character and extent of all surface and underground workings,
whether placer or lode, for mining purposes, locating and describing them,
as required by section 41.
(rf) The proximity of centres of trade or residence.
{e] The proximity of well-known systems of lode deposits or of in-
dividual lodes.
(0 The use or adaptability of the claim for placer mining, and whether
water has been brought upon it in sufficient quantity to mine the same, or
whether it can be procured for that purpose.
{g) What works or expenditures have been made by the claimant or
his grantors for the development of the claim, and their situation and
location with respect to the same as applied for.
{]%) The true situation of all mines, salt licks, salt springs, and mill
sites, which come to your knowledge, or report that none exist on the
claim, as the facts may warrant.
(0 Said report must be made under oath, and duly corroborated by
one or more disinterested persons.
57. Descriptive reports on placer claims taken by legal subdivisions
are authorized only by special order, and must contain a description of
the claim in addition to the foregoing requirements.
PRACTICE.
58. The practice of employing the claimants, their attorneys, or parties
in interest, as assistants in making surveys of mineral claims, will not be
allowed.
59. Your field work must be accurately and properly performed and
your returns made in conformity with the foregoing instructions. Errors
in the survey must be corrected at your own expense, and if the time re-
quired in the examination of your returns is increased by reison of your
neglect or carelessness, you will be required to make an additional deposit
for office work. You will be held to a strict accountability for the faithful
discharge of your duties, and will be required to observe fully the re-
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APPENDIX B. 655
quirements and regulations in force as to making mineral surveys. If
found incompetent as a surveyor, careless in the discharge of your duties,
or guilty of a violation of said regulations, your appointment will be
promptly revoked.
APPLICATION TO UNITED STATES SURVEYOR-GENERAL
FOR SURVEY OF .MINING CLAIM.
(4-689.)
Denver, Colo., January 25, 1893.
United States Surveyor-general,
Denver, Colorado.
Sir: T. E. Jenkins et al., claimants, hereby make application for an
official survey, under the provisions of Chapter Six, Title Thirty-two,
of the Revised Statutes of the United States, and regulations and in-
structions thereunder, of the mining claim known as the Cumro Placer
and Poorman, Hawley, ^tna, and Podunk lodes and the Poorman mill
site, situate in Pike's Peak Mining District, El Paso County, Colorado,
in Sections 17, 19, and 20, Township No. 14 S., Range No. 69 W. Said
claim is based upon valid locations made on various dates, 18 , and
duly recorded on various dates, 18 , and is fully described in the duly
certified copies of the record of the location certificates filed herewith.
Said certificates contain the name of the locators, the dates of location,
and such a definite description of the claim by reference to natural ob-
jects or permanent monuments as will identify the claim, and said loca-
tions have been distinctly marked by monuments on the ground, so that
their boundaries can be readily traced.
I request that you will send me an estimate of the amount required
to defray the expenses of platting and other work in your office, re-
quired under the regulations, that I may make proper deposit therefor,
and that thereupon you will cause the survey to be made by A. L.
Hawley, United States deputy mineral surveyor, and proper action to be
taken thereon by your office, as required by the United States mining
laws and regulations thereunder.
T. E. Jenkins,
For himself and co-claimants.
P. O. address: Denver,
Arapahoe County, Colorado.
(4-682.)
ORDER FOR MINERAL SURVEY.
Department of the Interior,
Office of U. S. Surveyor-General,
Denver, Colo., February 6, 1893.
To A. L. Hawley,
U. S. Deputy Mineral Surveyor,
Denver, Colorado.
Sir: Application has been filed in this office by T. E. Jenkins et al.,
dated January 25, 1893, for an official survey of the mining claim of T. E.
Jenkins et aL, known as the Cumro Placer and Poorman, Hawley, ^tna,
and Podunk lodes and Poorman mill site, situate in Pike's Peak Mining
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656 SUkVMVmC,
District, El Paso County, in Sections 17, 19, and 20, Township No. 14 S.,
Range No. 69 W., which claim is based upon locations made on May i,
1892, May 4, 1892, May 4, 1888. June 4, 1892, June 14. 1891, and Decem-
ber 5, 1891, respectively, and duly recorded various dates, 18 , and is
fully described in the duly certified copies of the record of the location
certificates filed by the apphcants for said survey, copies of which are
herewith inclosed. You are hereby directed to make the survey of said
claim in strict conformity with existing laws, official regulations, and in-
structions thereunder, and to make proper return to this office. Said sur-
vey will be designated as Survey No. 8000, A and B.
Very respectfully,
U. S. Surveyor-General
for Colorado.
SPECIMEN FIELD NOTES.
(4-683.^
Mineral Survey No 8000, A and B . . . .
Lot No
Pueblo Land District.
FIELD NOTES
OF THE SURVEY OF THE MINING CLAIM OF
J, T. E. Jenkins, et al.,
KNOWN AS THE
.... Cumro Placer and Poorman, Hawley. ^tna, and Podunk Lodes,
and Poorman M ill Site
Pike's Peak Mining District,
El Paso County, Colorado
Sections 17, 19, and 20, Township 14 S.,
Range 69 W
Surveyed under instructions dated February 6, 1893
by A. L. HAWLEY,
U. S. Deputy Mineral Surveyor,
Claim located , 18. . . .
Survey commenced February 9, 1893. . . .
Survey completed February 12, 1893
Address of claimants:
Denver,
Colorado. . . .
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APPENDIX B.
657
Fmt.
660.
182.3
84.5
67.2
SURVEY 8000 A.
CUMRO PLACER.
TRACT A.
Beginning at Cor. No. i.
Identical with the SW. Cor. of the location and with the
SW. Cor. of sec. 17, T. 14 S., R. 69 W. of the 6th Principal
Meridian.
A pine post, ^Yz ft. long, 4 ins. square, set 2 ft. in the ground,
with mound of stone, alongside the section corner, scribed
1-8000 A, whence
A spruce, 17 ins. diam., bears N. 8* 41' W. 7 ft., and a spruce,
14 ins. diam., bears S. 68° 14', E. 18.5 ft., each blazed and
scribed B. T. 1-8000 A.
James Peak bears N. 52** 21' W.
Hahns Peak bears N. 29° 28' W.
Thence North.
Va. 14° 22' E.
To Cor. No. 2.
Identical with a corner of the location.
A pine post, ^Yi ft. long, 4 ins. square, set 18 ins. in the
^^round, with mound of earth and stone, scribed 2-8000 A,
whence
A spruce, 18 ins. diam., blazed and scribed B. T. 2-8000 A,
bears S. 14** 47', E. 17.3 ft.
Thence N. 89** 50' E.
Va. 14° 28' E.
To Cor. No. 3.
On line 1-2 Hawley lode of this survey,
A cedar post, 5 ft. long, 4 ins. square, set 2 ft in the ground,
with mound of earth, scribed 3-8000 A, whence
A corner of the location bears N. 89° 50' E. 1 126.7 ft.
Thence S. 3^ 48' E.
Va. 14' 28' E.
To Cor. No. 4.
A pine post, 4Y2 ft. long, 4 ins. square, set 2 ft. in the ground,
with mound of stone, scribed 4-1-8000 A, whence
A pine, 14 ins. diam., bears S. 21° 47', E. 14.3 ft., and a
spruce, II ins. diam., bears N. 14° 52' E. 6 ft., each blazed and
scribed B. T. 4-1-8000 A.
Thence N. 86** 12' E.
Va. 14° 28' E.
To Cor. No. 5-
On line 1-2 Poorman lode of this survey.
A Cottonwood post, 5 ft. long, 4 ins. square, set 2 ft. in tho
ground, with mound of stone, scribed 5-8000 A.
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658
SURVEYING.
Feet.
243.4
350.48
542.38
416.8
555.44
1329.42
Thence S. 17° 38' W.
Va. 14° 36' E.
To Cor. No, 6.
A pine post, aVi ft. long, 4 ins. square, set 2 ft. in the groun(\
with mound of earth, scribed 6-1-8000 A, whence
A high Mt. bears N. 51° 14' E.
Thence S. 41° 14' E.
Va. 14° 30' E.
To Cor. No. 7.
A granite bowlder, 27 x 12 x 9 ins., set 16 ins. in the ground,
chiselled 7-6-8000 A.
Thence N. 17° 38' E.
Va. 14° 30' E.
To Cor. No. 8.
On line 4-1 Hawley lode of this survey.
A pine post, 4J^ ft. long, 5 ins. square, set 2 ft. in the ground,
scribed 8-8000 A.
Thence N. 86° 12' E.
Va. 14° 28' E.
To Cor. No. 9.
On line 3-4 Podunk lode of this survey.
A granite stone, 26 x 16 x 6 ins., set 18 ins. in the ground,
with mound of stone, chiselled 9-8000 A.
Thence S. 41° 14' E.
Va. 14° 28' E.
To Cor. No. 10.
On line 4-1 Sur. No. 7000, Ajax lode, claimant unknown.
A pine post, 4-1 ft. long, 4 ins. square, set 18 ins. in the
ground, with mound of earth, scribed 10-8000 A.
Thence S. 7° 45' W.
Va. 14° 25' E.
To Cor. No. II.
On line 4-1 Sur. No. 7000, Ajax loc, at N. 75° 45' E., 10.73
ft. from Cor. No. 4.
A pine post, 5 t. long, 4 ins. square, set 2 ft. in the ground,
with mound of earth and stone, scribed 1 1-8000 A, whence
A Cottonwood post, 8 ins. diam., blazed and scribed B. T.
11-8000 A, bears N. zz'' 27' W. 5 ft.
Thence S. 89° 50'
Va. 14° 25' E.
To Cor. No. I, the place of beginning.
W.
TRACT B.
Beginning at Cor. No. 12.
At intersection of lines 3-4, ^Etna lode of this survey, and
Aztec lode, unsurveyed, John Doe, claimant.
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APPENDIX B.
659
Feet.
397.33
300.
330.7
769.6
265.
758.1
819. 1
A spruce post, 4^ ft. long, 4 ins. square, set 18 ins. in the
ground, scribed 12—8000 A, whence
A pine stump, 18 ins. diam., 3 ft. high, blazed and scribed
B. S. ia-8ooo A, bears N. 89^ 11' E. 9.4 ft.
Thence S. 41° 14' E.
Va. 14° 30' E.
To Cor. No. 13.
A sandstone, 36x20x4 ins., set 16 ins. in the ground with
mound of stone, chiselled 13-3-8000 A.
Thence S. 48° 46' W.
Va. 14° 30' E.
Cor. Nos. 2, i^tna and Podunk lodes of this survey, a point in
Cumro Creek, 4 ft. wide, flows west.
To Cor. No. 14.
A rock in place 6x4x2 ft. above the general surface,
chiselled cross (x) at corner point and 14— W. C. 2-2-8000 A,
whence
Cor. No. II, Tract A, of this survey, bears S. 89° 50' W. 539
ft.
Thence N. 89° 50' E.
Va. 14° 30' E.
To Cor. No. 15.
Identical with the S. J4 Cor. of said Sec. 17, and with the
SE. Cor. of the location.
A granite stone, 12 x 10 x 6 ins. above the ground, chiselled
15-8000 A.
Thence North.
Va. 14° 30' E.
Cumro Creek, 4 ft. wide, flows S. 65° W.
To Cor. No. 16.
On line 3-4, Aztec lode, unsurveyed, at S. 72" 43' W. 115.6
ft. from Cor. No. 3.
A spruce post, 45^ ft. long, 4 ins. square, set 6 ins. in the
ground to bed rock, with mound of earth and stone, scribed
16-8000 A.
Thence S. ^2'' 43' W.
Va. 14° 28' E.
To Cor. No. 12, the place of beginning.
TRACT c.
Beginning at Cor. No. 17.
On line 1-2, Aztec lode, unsurveyed, at S. ^2^ 43' W. 22.26
ft. from Cor. No. 2.
A pine post, 45^ ft. long, 4 ins. square, set 12 ins. in the
ground to bed rock, with mound of stone, scribed 17-8000 A,
whence
Cor. No. 16, Tract B, of this survey, bears S. 314.2 ft.
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66o
PURVEYING,
Thence S. 72** 43' W.
Va. lA"" 40' E.
To Cor. No. 18.
At intersection of lines 1—2 and 2—3, Aztec lode, unsurveyed,
and Sur. No. 7000, Ajax lode, respectively.
A cross (x^ at corner point and 18-8000 A chiselled on a
granite rock in place, showing 10 x 3 x 2 ft. above the general
surface.
Thence N. 7° 45' E.
Va. 14° 30' E.
To Cor. No. 19.
On line 2-3, Sur. No. 7000, Ajax lode, at S. 7** 45' W. 116.4
ft. from Cor. No. 2.
A granite stone, 28 x 10 x 3 ins., set 12 ins. in the ground,
chiselled 19-8005, whence
A corner of the location bears S. 89° 50' W. 484.6 ft.
Thence N. 89° 50' E.
Va. 14^* 30' E.
To Cor. No. 20.
A granite rock, 30 x 20 x 16 ins., set 16 ins. in the ground,
chiselled 20-8000 A, whence
A cross (x) and B. R. 20-8000 A, chiselled 4 ft. above the
ground on a limestone cliff 20 ft. high, bears S. 83° ii' E.
45.6 ft
Thence South.
Va. 14° 30' E.
To Cor. No. 17, the place of beginning.
AREA.
Tract A, containing 14.660 acres.
Tract B, containing 9.858 "
Tract C, containing 7-532 **
Total area Cumro placer 32.070 acres.
POORMAN LODE.
Beginning at Cor. No. i.
Identical with Cor. No. 6 Cumro placer of this survey,
whence
The SW. Cor. Sec. 17, T. 14 S., R. 69 W. of the 6th P. M.,
bears S. 27° 28' W. 39326 ft.
Thence N. 17** 38' E.
Cor. No. 5 Cumro placer and intersect line 4-1 Hawley lode,
both of this survey.
Intersect line 2-2, Hawley lode of this survey.
To Cor. No. 2.
A granite rock, 30 x 20 x 16 ins., set 16 ins. in the ground,
chiselled 2-8000 A.
Feet.
938.26
530.
824.43
247.72
243.4
565.7
831.4
Digitized byVjOOQlC
APPENDIX B,
66l
Feet.
661.57
300.
300.
578.
64.4
386.7
929.04
350.48
84.5
300.
185.
507.3
Thence N. 48^ 46' E.
To Cor. No. 3.
A pine post, 5 ft. long, 4 ins. square, set 22 ins. in the ground,
to bed rock, scribed 3-8000 A.
Thence S. 41° 14' E.
To Cor. No. 4.
A cedar post, 4^ ft. long, 5 ins. square, set 18 ins. in the
ground, scribed 4-4-8000 A, whence
Cor. No. I Sur. No. 7000, Ajax lode, bears N. 72° 22' E.
422.6 ft.
Thence S. 48° 46' W.
Cor. Nos. I ^tna and Podunk lodes of this survey.
To Cor. No. 5.
A cedar stump, 3 ft. high, hewed to 4 ins. square, surrounded
by mound of stone, scribed 5-4-8000 A, whence
A cross (x) and B. R. 5-4-8000 chiselled on a porphyry
stone, showing 9x6x4 ft. above the ground, bears N. 75° 14'
E. 2^.y ft.
Thence S. \t 38' W.
Intersect line 2—3 Hawley lode of this survey.
Cor. No. 8 Cumro placer and intersect line 4-1 Hawley lode,
both of this survey.
To Cor. No. 6.
Identical with Cor. No. 7 Cumro placer of this survey.
To Cor. No.
Thence N. 41^ 14' W.
I, the place of beginning.
HAWLEY LODE.
Beginning at Cor. No. i.
Identical with Cor. No. i of the location and with Cor.
4 Cumro placer of this survey, whence
The SW. Cor. Sec. 17, T. 14 S., R. 69 W. of the 6th P
bears S. 18° 4' W. 606.1 ft.
Cor. No. I Poorman lode of this survey bears S. 1° 41' W
227.6 ft.
No.
M.
Thence N. 3" 48' W.
Va. 14°. 28' E.
Cor. No. 3 Cumro placer of this survey.
To Cor. No. 2.
A sandstone, 30 x 12 x 2 ins., set 14 ins. in the ground,
chiselled 2-8000 A, whence
Cor. No. 4 of the location bears N. 45° W. 28.5 ft.
E.
Thence N. 86° 12'
Va. 14° 25' E.
Intersect line 1-2 Poorman lode of this survey.
Intersect line 5-6 Poorman lode of this survey.
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662
SURVEYING,
Feet.
567.6
926.7
1264.7
150.
250.6
300.
30.8
108.3
206.
458.4
875.2
1197.5
1264.7
288.3
641.2
Intersect line 3-4 Podunk lode of this survey.
Intersect lines i— 2 Podunk and ^tna lodes of this survey.
To Cor. No. 3.
A granite stone, 26x14x8 ins., set 12 ins. in the ground
with mound of stone, chiselled 3-8000 A, whence
Cor. No. 3 of the location bears N. 86° 12' E. 235.3 ^t.
Cor. No. I Sur. No. 7000, Ajax lode, bears N. ii° E. 529.9 ft.
Cor. No. I Aztec lode, unsurveyed, bears S. 12"" 30' W.
378.4 ft.
Thence S. 3° 48' E.
Va. 14° 2^' E.
Intersect line 4-1 Sur. No. 7000, Ajax lode, at S. 7° 45' W.
676 ft. from Cor. No. i.
Intersect line 1-2 Aztec lode, unsurveyed, at N. 72° 43' E.
229.8 ft. from Cor. No. i.
To Cor. No. 4.
A pine post, ^Yi ft. long, 4 ins. square, set 18 ins. in the
ground, scribed 4—8000 A, whence
Cor. No. 4 of the location bears N. 86** 12' E. 235.3 ft.
Thence S. 86° 12' W.
Va. 1^" 15' E.
Intersect line 4-1 Sur. No. 7000, Ajax lode, at S. 7° 45' W.
829.1 ft. from Cor. No. i.
Intersect lines i— 2 ^tna and Podunk lodes of this survey.
Intersect line 1-2 Aztec lode, unsurveyed, at N. '^2° 43' E. 18. i
ft. from Cor. No. i.
Cor. No. 9 Cumro placer and intersect line 3—4 Podunk lode,
both of this survey.
Cor. No. 8 Cumro placer and intersect line 5-6 Poorman lode,
both of this survey.
Cor. No. 5 Cumro placer and intersect line 1-2 Poorman lode,
both of this survey.
To Cor. No. I, the place of beginning.
^TNA LODE.
Beginning at Cor. No. i.
On line 4-5 Poorman lode of this survey.
A pine post, 4^ ft. long, 4 ins. square, set 18 ins. in the
ground, scribed i— 1-8000 A, whence
The SW. Cor. Sec. 17, T. 14 S., R. 69 W. of the 6th P. M.,
bears S. 38° 2' VV. 1465 ft.
Cor. No. I Aztec lode, unsurveyed, bears S. 32' 19' E. 607.76
ft.
Cor. No. 3 Hawley lode of this survey, bears S. 69° 46' E.
561.9 ft.
Thence S. 41° 14' E.
Intersect line 2-3 Hawley lode of this survey.
Intersect line 1-2 Aztec lode, unsurveyed, at N. 72** 43' E. 103.1
ft. from Cor. No i.
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APPENDIX B.
663
Feet.
666.1
766.9
969.5
1 164. 7
1500.
300.
397.33
596.43
725.6
994.
1500.
300.
288.3
641.2
666.1
766.9
969.5
Intersect line 4-1 Hawley lode of this survey.
Intersect line 4-1 Sur. No. 7000. Ajax lode, at S. 7° 45' W.
910.9 ft. from Cor. No. i.
Intersect line 3-4 Aztec lode, unsurvcyed, at N. •ji'' 43' E. 236.3
ft. from Cor. No. 4.
Intersect line 2-1 Sur. No. 7000, Ajax lode, at N. 7° 45' E.
328.2 ft. from Cor. No. 3.
To Cor. No. 2.
On line 13—14 Cumro placer of this survey.
Not set, as it falls in the centre of Cumro Creek, wliere per-
manent corner could not be estab'ished, whence
Witness corner to Cor. No. 2. identical with Cor. No. 14
Cumro placer of this survey, bears S. 48° /16' W. 30.7 ft.
Cor. No. 3 Sur. No. 7000, Ajax lode, bears S. 74° 38' W.
275.2 ft.
Thence N. 48° 46' E.
To Cor. No. 3.
Identical with Cor. No. 13 Cumro i acer of this survey.
Thence N. 41' 14' W.
Cor. No. 12 Cumro placer of this survey and intersect line
3-4 Aztec lode, unsurveyed, at N. ^2"* 43' E. 564.6 ft. from
Cor. No. 4.
Intersect line z-^^ Sur. No. 7000, Ajax lode, at N. 7° 45' E. 725.8
ft. from Cor. No. 3.
Intersect line i— 2 Aztec lode, unsurveyed, at N. 72° 43' E. 431.3
ft. from Cor. No. i.
Intersect line 4-1 Sur. No. 7000, Ajax lode, at S. 7° 45' W.
513.2 ft. from Cor. No. i.
To Cor. No. 4.
Identical with Cor. No. 4 Poorman lode of this survey.
Thence S. 48° 46' W.
To Cor. No. I, the place of beginning.
PODUNK LODE.
Beginning at Cor. No. i.
Identical with Cor. No. i ^tna lode of this survey, whence
Cor. No. 2 of the location bears N. 48^* 46' E. 22 ft.
The SW. Cor. Sec. 17. T. 14 S., R. 69 W. of the 6th P. M.,
bears S. 38° 2' W. 1465 ft.
Thence S. 41° 14' E.
Intersect line 2—3 Hawley lode of this survey.
Intersect line 1-2 Aztec lode, unsurveyed, at N. ^2° 43' E. 103. i
ft. from Cor. No. i.
Intersect line 4-1 Hawley lode of tliis survey.
Intersect line 4-1 Sur. No. 7000, Ajax lode, at S. 7** 45' W.
910.9 ft. from Cor. No. i.
Intersect line z-A Aztec lode, unsurveyed, at N. ^2^ 43' E.
236.3 ft. from Cor. No. 4.
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664
SURVEYING,
Feet.
I 164. 7
1500.
I
30.7
250.
278.
37.
155.
491.3
1046.7
1424.5
1500.
278.
Intersect line 2-3 Sur. No. 7000, Ajax lode, at N. 7** 45' E.
328.2 ft. from Cor. No. 3.
To Cor. No. 2.
In Cumro creek.
Identical with Cor. No. 2 /Etna lode of this survey, whence
Cor. No. 3 of the location bears N. 48° 46' E. 22 ft.
Thence S. 48° 46' W.
Witness corner to Cor. No. 2.
Identical with witness corner to Cor. No. 2 ^tna lode, and
with Cor. No. 14 Cumro placer, both of this survey.
Witness corner to Cor. No. 3.
A granite stone, 30x20x4 ins., set 14 ins. in the ground,
with mound of stone, chiselled W. C. 3-8000 A, whence
A cedar stump. 14 ins. diam., 2 ft. high, blazed and scribed
B. S. W. C. 3-8000 A, bears N. 7° 56' W. 8.4 ft.
Pike's Peak bears N. 5° E.
To Cor. No. 3.
On face of inaccessible granite cliff.
Identical with Cor. 4 of the location.
Thence N. 41° 14' W.
Cumro creek, 4 ft. wide, course S. 80** W.
Intersect line 3-4 Sur. No. 7000, Ajax lode, at N. 82** 15' W.
46.3 ft. from Cor. No. 3.
Cor. No. 10 Cumro placer of this survey and intersect line 4—1.
Sur. No. 7000, Ajax lode, at N. 7° 45' E. 220.7 ^t. from Cor.
No. 4.
Cor. No. 9 Cumro placer and intersect line 4-1 Hawley lode,
both of this survey.
Intersect line 2-2t Hawley lode of this survey.
To Cor. No. 4.
Identical with Cor. No. 5 Poorman lode of this survey and
with Cor. No. i of the location.
Thence N. 48° 46' E.
To Cor. No. I, the place of beginning.
Variation at all corners of the Poorman, i^tna, and Podunk
lodes, 14° 30' E.
AREA.
Total area Hawley lode 8.710 acres.
Area in conflict with
Poorman lode of this survey 2.220 "
Sur. No. 7000. Ajax lode 053 **
Aztec lode, unsurveyed 117 "
Aztec lode, unsurveyed (exclusive of its conflict with
Sur. /ooo, Ajax lode) 089 "
Total area Hawley lode S.7jq acres.
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APPENDIX B. 665
Less area in conflict with
Poorman lode of this survey 2.220 acres.
Sur. No. 7000, Ajax lode 053 "
Aztec lode, unsurveyed 089 " = 2.362 acres.
Net area Hawley lode 6.348 acres.
Total area ^tna lode 10.331 acres.
Area in conflict with
Hawley lode of this survey 1.537 "
Sur. No. 7000, Ajax lode 2.738 "
Sur. No. 7000, Ajax lode (exclusive of its conflict
with Hawley lode of this survey) 2.167 "
Aztec lode, unsurveyed 2.261 "
Aztec lode, unsurveyed (exclusive of its conflict
with Hawley lode of this survey) 2.167 "
Aztec lode, unsurveyed (exclusive of its conflict
with Sur. No. 7000, Ajax lode) 484 "
Aztec lode, unsurveyed (exclusive of its conflict
with Hawley lode of this survey and Sur. No.
7000, Ajax lode) 418 "
Total area ^Etna lode 10.331 acres.
Less area in conflict with
Hawley lode of this survey 1-537 acres.
Sur. No. 7000. Ajax lode 2.685 "
Aztec lode, unsurveyed 418 " = 4.640 "
Net area i^tna lode 5.691 acres.
Total area Podunk lode 9-573 acres.
Area in conflict with
Hawley lode of this survey 2.41 1 "
Sur. No. 7000, Ajax lode 2.526 "
Aztec lode, unsurveyed 1.169 "
Aztec lode, unsurveyed (exclusive of its conflict with
Hawley lode of this survey) i-i47 "
Aztec lode, unsurveyed (exclusive of its conflict with
Sur. No. 7000, Ajax lode) 8ii "
Aztec lode, unsurveyed (exclusive of its conflict with
Hawley lode of this survey and Sur. No. 7000,
Ajax lode) 789 "
Total area Podunk lode 9-573 acres.
Less area in conflict with
Hawley lode of this survey 2.41 1 acres.
Sur. No. 7000, Ajax lode 2.516 **
Aztec lode, unsurveyed 789 *' =5.716 "
Net area Podunk lode 3857 acres.
Net area Haw'ey lode 6.348 "
Net area i^tna lode 5.691 "
Total and net area Poorman lode 10.3JI *'
Net area lode claim 26.227 acres.
Digitized byVjOOQlC
666
SURVEYING,
Feet.
65.
858.
921.2
467.66
700.
SURVEY NO. 8000 B.
POOR.\fAN MILL SITE.
Beginning at Cor. No. i.
A spruce post, 5 ft. long, 4 ins. square, set 18 ins. in the
ground, scribed 1-8000 B, whence
Cor. No. 6 Sur. No. 8000 A, Poorman lode, bears N. 50° 8'
E. 3782 ft.
The NE. Cor. Sec. 19, T. 14 S., R. 69 W. oi the 6th P. M..
bears N. 46° 48' E. 3416.9 ft.
Thence S. 85" 51' W.
Cumro creeiv, 4 ft. wide, flows N. 65** W.
Cumro creek, 4 ft. wide, flows South.
To Cor. No. 2.
A granite stone, 28 x 12 x 10 ins., set 12 ins. in the ground,
with mound of stone, chiselled 2-8000 B, whence
A pine, 12 ins. diam., blazed and scribed B. T. 2-8000 B,
bears W. 9.5 ft.
Thence N. zf 55' E.
To Cor. No. 3.
A granite stone, 30 x 12 x 8 ins., set 18 ins. in the ground,
with mound of stone, chiselled 3-8000 B.
Thence S. 64° 25' E.
To Cor. No. I, the place of beginning.
Containing 3.671 acres.
Variation at all corners, 14** 45' E.
The surveys of the Poorman and ^Tltna lodes and the Poorman mill
site are identical with the respective locations.
LOCATION.
This claim is located in the SW. J4 of Sec. 17, the NE. J4 of Sec.
and the NW. V* of Sec. 20, T. 14 S., R. 69 W. of the 6th P. M.
19.
EXPENDITURE OF FIVE HUNDRED DOLLARS.
I certify that the value of the labor and improvements upon this claim,
placed thereon by the claimants and their grantors, is not less than five
hundred dollars, and that said improvements consist of:
Placer workings, the centre of the northeasterly end of which bears
from Cor. No. 15, Cumro placer. N. 46** W. 285 ft., averaging 40 ft. wide
and 8 ft. deep, and extending S. 62" W. 120 ft. along the bed of Cumro
creek. Value, $800.
The discovery shaft of the Poorman lode, which bears from Cor. No. 5
N. $6'' 48' W. 1557 ^t» (>x4 ft., 12 ft. deep. Value, $ioo.
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PLATE V.
TRANSVCRSe SECTION
NORTH AND SOUTH
MAP OF
THE LINCOLN MINE
JEFFERSON COUNTY, COLORADO
•URVEvio lY ROtorr t. rrooKTON, iirr. lom ism
OMQINALtOALl: 100 FT. TO I INOH
MALI OP mOUOIO PIATC .* 1« FT. TO I MON
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I
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BPECIAISN BUBVR OV lONIKO GLAOI.
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SURVEY No. 8000 A. & B.
PUEBLO — LAND DISTRICT
SURVEYED FEBRUARY »-12, 1893
by, A.LHamlni
(/ JL Dt^ Him.
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APPENDIX B. 667
The discovery cut of the Podunk lode, the face of which bears from
Cor. No. 4 S. 88° 13' E. 205.2 ft, 5 ft. wide, 10 ft. face, running N. 50**
E. 24 ft. in earth and rock. Value, $110.
The incline discovery shaft of the ^^tna lode, the mouth of which
is on the centre line 75 ft. from the centre of line 4— i, 4^x6 ft, 24 ft
deep in rock, timbered, course S. 48°, dip. 60°. Value, $250.
The last 120 ft. of a tunnel, the mouth of which bears from Cor. No. 6,
Poorman lode, N. 67° 48' E. 582 ft, 5x6 ft., running N. lo** 44' W. 515 ft
to breast. The point cf discovery of the Hawley lode is in this tunnel
475 ft. from the mouth, and bears from Cor. No. i N. 75° 52' E. 702.5 ft.
Value of last 120 ft., $2,300.
This tunnel is in course of construction for the development of this
claim and Surs. Nos. 6582 and 6583, Roy and Raymond lodes, claimants
-unknown. An undivided half interest in the first 375 ft. of this tunnel has
been credited to each of the last two mentioned surveys.
A shaft on the centre line of the Hawley lode 672 ft. from the centre
of line 1-2, 3V^ x 6 ft., 20 ft. deep in earth and rock, timbered, at the bot-
tom, of which is a drift 4x6 ft. running N. 86° E. 18 ft.
Value of shaft and drift, $300.
The surface embraced by this claim ascends rapidly from the mouth
yf the tunnel towards Cor. No. 3 of the Poorman lode, the northerly ends
of the Poorman, ^tna, and Podunk lodes being from 300 to 500 ft. higher
than the mouth of the tunnel. The veins of the v?£tna and Podunk lodes
dip about 60° to the SW. The tunnel, described and included in the esti-
mate of expenditure, continued in its present course, will cut the veins of
the several locations at great depth, whereby by one system of workings
and one plant of machinery the entire claim can be most advantageously
«nd economically developed.
OTHER IMPROVEMENTS.
A shaft which bears from Cor. No. 20 Cumro placer, S. 48° 30' W.
305 ft., 3x5 ft., 12 ft. deep in earth and gravel; A. K. Smith, claimant.
A log cabin, the W. corner of which bears from Cor. No. 13 Cumro
placer, S. 40° E. 120 ft., 12 x 16 ft., course of long sides N. 44° E.
A log cabin, the NW. Cor. of which bears from Cor. No. 3 Sur. No.
8000 B, Poorman mill site, S. 10° 40' E. 107 ft., 16 ft square, course of
sides S. 3° E. Said cabins belong to claimants herein.
INSTRUMENT.
This survey was made with a Gurley Light Mountain Transit The
courses were deflected from the true meridian as determined by direct
solar observations. The distances were measured with 50-ft. and 500-ft.
steel tapes.
Note. — The disagreements between these field notes and the location
certificates of the Hawley and i^tna lodes and the Poorman mill site,
with regard to the position of the discovery point and the course of the
boundary lines, are due to errors in the latter.
Note. — Neither E. E. Ames nor myself, who appear as locators of
the Hawley lode, held any interest, directly or indirectly, in this claim
at the time of making the survey, having sold our interests in June, 1890.
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6o8 SURVEYING.
(4-^5.)
FINAL OATHS FOR SURVEYS.
LIST OF NAMES.
A list of the names of the individuals employed by A. L.
Hawley , United States deputy mineral surveyor, to assist in running,
measuring and marking the lines, corners, and boundaries described in
the foregoing field notes of the survey of the mining claim of T. E,
Jenkins ct al , known as the Cumro placer and Poorman, Hawley,
-^tna, and Podunk lodes, and Poorman mill site. . . ., and showing the re-
spective capacities in which they acted:
, Chaintnan,
E. E. Ames , Chainman,
....G. W. Trommlitz , Axeman.
., Flagman.
FINAL OATHS OF ASSISTANTS.
We, E. E. Ames and G. W. Trommlitz do solemnly
swear that we assisted A. L. Hawley , United States deputy min-
eral sisrveyor. in marking the corners and surveying the boundaries of
the mining claim of T. E. Jenkins et al , known as the Cumro
placer and Poorman, Hawley, JEins., and Podunk lodes, and Poorman
mill site represented in the foregoing field notes as having been sur-
veyed by said deputy mineral surveyor and under his direction; and that
said survey has been in all respects, to the best of our knowledge and
belief, faithfully and correctly executed, and the corner and boundary
monuments established according to law and the instructions furnished
by the United States Surveyor-General for. .. .Colorado
, Chainman.
E. E. Ames , Chainmun.
G. W. Trommlitz , Axeman.
, Flagman,
Subscribed and sworn to by the above-named persons before me this
.... 13th day of February , 1893.
[seal.] A. L. Hawley ,
Notary Public ,
El Paso County f Colorado.
My commission expires July 28th, 1896.
(4-686.)
FINAL OATH OF UNITED STATES DEPUTY MINERAL
SURVEYOR.
I, A. L. Hawley , United States deputy mineral surveyor.
do solemnly swear that, in pursuance of instructions received from the
United States Surveyor-General for Colorado , dated February
6th, 1803 , I have, in strict conformity to the laws of the United States.
the official rep^ulations and instructions thereunder, and the instructions
of said Surveyor-General, faithfully and correctly executed the survey of
the Mining Claim of. . . .T. E. Jenkins ct al , known as the. . . .Cumro
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APPENDIX B. 669
placer and Poorman, Hawley, ^^tna, and Podunk lodes, and Poorman
mill site , situate in Pike'sPeak. .. .Mining District...., El Paso
.'...County , Colorado , in Sections 17, 19, and 20 , Township
No 14 S Range No 69 W. of the 6th P. M , and desig-
nated as Survey No 8000 A and B , as represented in the fore-
going field notes, which accurately show the boundaries of said mining
claim as distinctly marked by monuments on the ground, and described
in the attached copy of each location certificate, which was received by
me from the Surveyor-General with said instructions, and that all the
corners of said survey have been established and perpetuated in strict
accordance with the law, official regulations and instructions thereunder;
and I do further solemnly swear that the f jregoing are the true and
original field notes of said survey and my report therein, and that the
labor expended and improvements made upon said mining claim by
claimants or their grantors are as therein fully stated, and that
the character, extent, location, and itemized value thereof are specified
therein with particularity and full detail, and that no portion of said
labor or improvements so credited to this claim has been included in the
estimate of expenditures upon any other claim.
A. L. HawleV ,
United States Deputy Mineral Surveyor.
Subscribed and sworn to by the said A. L. Hawley United
States deputy mineral surveyor, before me a Justice of the Peace in
and for El Paso County, Colorado , this.... 20th day of Feb-
ruary , 1893.
Geo. K. Kimball ,
Justice of the Peace.
LOCATION CERTIFICATE— PLACER CLAIM.
Know all men by these presents. That I, T. E. Jenkins, the un-
dersigned citizen of the United States, resident of the County of Arapahoe
and State of Colorado, having complied with the provisions of Chap-
ter 6, Title XXXII, of the Revised Statutes of the United States, and
with local customs, laws, and regulations, claim by right of discovery and
location, as a placer claim, the following-described premises, situate, lying
and being in Pike's Peak mining district, County of El Paso, and State of
Colorado, to wit:
The SE. M of the SW. ^, and the S. y2 of the SW. H of the SW.
54 of Sec. 17, T. 14 S., R. 69 W. of the 6th P. M. To be known as the
CuMRO Placer.
Located May ist, 1892.
Date of certificate, June 4th, 1892.
T. E. Jenkins.
LOCATION CERTIFICATE— LODE CLAIM.
State of Colorado, )
County of.. El Paso.., S
Know all men by these presents, That O. F. Shattuck ,
the undersigned, has this 4th day of May , 1892, located and
claimed, and by these presents does locate and claim, by right of dis-
covery and location, in compliance with the mining acts of Congress,
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670 SURVEYING.
approved May 10, 1872, and all subsequent acts, and with local customs,
laws, and regulations 1500 linear feet and horizontal measure-
ment on the Poorman lode, vein, ledge, or deposit, along the vein
thereof, with all its dips, angles, and variations, as allowed by law, to-
gether with 150 feet on the westerly .... side and 150 feet
on the easterly side of the middle of said vein at the surface, so
far as can be determined from present developments; and all veins, lodes,
ledges, or deposits and surface ground within the lines of said claim,
620 feet running N. 48** 46' E.^ . . .from centre of discovery
shaft and 880 feet running S. 17** 38' W .from centre of
discovery shaft ; said discovery shaft being situate upon said
lode, vein, ledge, or deposit, and within the lines of said claim, in
Pike's Peak Mining District, County of El Paso.... and State of
Colorado, described by metes and bounds as follows, to wit:
Beginning at Corner No. i Whence Cor. to Sees. 17, 18, 19, and
20, T. 14 S., R. 69 W. bears S. 27** 28' E. 393.26 ft., thence N. \f 38' E,
831.34 ft. to Cor. No. 2, thence N. 48° 46' E. 661.7 ft. to Cor. No. 3. thence
S. 41° 14' E. 300 ft. to Cor. No. 4, thence S. 48** 46' W. 578 ft. to Cor. No.
5, thence S. 17° 38' W. 929.04 ft. to Cor. No. 6, thence N. 41** 14' W. 350.48
ft. to Cor. No. I, the place of beginning
. ..O. F. Shattuck.
SBAL.
SEAU]
'SKAL.'
'SBAL.]
SKAL.]
'SRAL.'
SKAL.]
Said lode was discovered on the lath....
day of.... April...., A.D. 1893.
Attest:
Date of location May 4th, A.D.
1893.
Date of certificate June 1st, A.D.
189a. J
AMENDED LOCATION CERTIFICATE— LAW OF 1889.
State of Colorado, )
County of,. El Paso.., ]
Know all men by these presents. That A. L. Hawley and E. E.
Ames , the undersigned, have this 4th.... day of May , 1888,
amended, located, and claimed, and by these presents do amend, locate,
and claim, by right of discovery and amended location, in compliance
with the mining acts of Congress, approved May 10, 1872, and all subse-
quent acts, and with section 2409 of the general laws of Colorado, and
with local customs, laws, and regulations.. . .1500 linear feet and
horizontal measurement, on the Ilawley lode, vein, ledge, or de-
posit, alonfj the vein thereof, with all its dips, angles, and variations, as
allowed by law, together with 150 feet on each side of the middle of
said vein at the surface, so far as can be determined from present de-
velopments, and all veins, lodes, ledges, or deposits and surface ground
within the lines of said claim 8do feet running easterly from
centre of discovery p;jnt in tunnel/s., and 700 feet running
....westerly from centre of discovery point , said discovery
point being situate upon said lode, vein, ledge, or deposit, and
within the lines of said claim in Pike's Peak Mining District, County
of El Paso and State of Colorado, described by metes and bounds
as follows, to wit:
Beginning at Corner No. i whence Cor. to Sees. 17, 18, 19, and
20, T. IS S., R. 69 W. bears S. 18° 4' E. 606.1 ft., thence N. 3** 48' E. 300
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APPENDIX B,
671
ft. to Cor. No. 2, thence N. 87° E. 1500 ft. to Cor. No. 3, thence S. 3° 48'
W. 300 ft. to Cor. No. 4, thence S. 87° W. 1500 ft. to Cor. No. i, the place
of beginning This being the same lode originally located on the
17th day of September , 1886, and recorded on the 12th....
day of December , 1886, in Book.... 4 , Page 48 in the
office of the Recorder of El Paso County. This further and
amended certificate of location is made without waiver of any previously
acquired rights, but for the purpose of correcting any errors in the original
location, description, or record
Said lode was discovered the.... 12th....
day of.... Sept , A.D. 1886.
Attest:
Date of nmended location May... 4tli
A.D. 1888.
Date oif amended certificate July 14th
, AD. 1888.
.A. L. Hawlky.
...E. E. Ames..
SBAL.l
SEAL.^
'SRAI..
'SKAI..
•SEAL.
S8AI..
SEAL.^
' LOCATION CERTIFICATE— LODE CLAIM.
State of Colorado, \
County of.. El Paso.., \^^-
Know all men by these presents, That.. . .Grant Safely , the
undersigned, has this 4th day of June , 1892, located and
claimed and by these presents does locate and claim, by right of dis-
covery and location, in compliance with the mining acts of Congress,
approved May 10, 1872, and all subsequent acts, and with local customs,
laws, and regulations 1500 linear feet and horizontal measure-
ments on the yEtna lode, vein, ledge, or deposit, along the vein
thereof, with all its dips, angles, and variations, as allowed by law, to-
gether with 150 feet on the southwesterly side, and 150
....feet on the northeasterly side of the middle of said vein at the
surface, so far as can be determined from present developments; and all
veins, lodes, ledges, or deposits and surface ground within the lines of
said claim. .. .75.. . .feet running N. 40° W from centre of dis-
covery cut and 1425 feet running S. 40° E from centre of
discovery cut ; said discovery cut being situate upon said lode
vein, ledge, or deposit, and within the lines of said claim in Pike's
Peak Mining District, County of El Paso and State of Colo-
rado, described by metes and bounds as follows, to wit:
Beginning at Corner No. i whence SW. Cor. Sec. 17, T. 15 S., R.
69 W. bears S. 38° 2' E. 1465 ft., thence S. 40° E., 1500 ft. to Cor. No. 2,
thence N. 50° E. 300 ft. to Cor. No. 3, thence N. 40° W. 1500 ft. to Cor.
No. 4, thence S. 50° W. 300 ft. to Cor. No. i, the place of beginning
Said lode was discovered on the....ist
day of May...., A.D. 189a.
Attest:
Date of location June 4th , A.D. >•
189a.
Date of certificate July 6th
X«02.
.Grant Ga: : ly
.,A.D.
SKAI.
SE.-.L.
SEAL.
SEAL.
^SBAL.
SKAL.
SEAL.
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6/2 SURVEYING,
ADDITIONAL AND AMENDED LOCATION CERTIFICATE.
Law of 1889.
State of Colorado, /
County of, .El Paso. ., ) ^^'
Know all men dy these presents, That John H. Routt , the
undersigned, has this 4th.... day of May , 1891, amended, lo-
cated, and claimed, and by these presents does amend, locate, and claim,
by right of the original discovery and this additional and amended loca-
tion certificate, in compliance with the mining acts of Congress, approved
May 10, 1872, and all subsequent acts, and with section 2409 of the Gen-
eral Statutes of Colorado, and with local customs, laws, and regulations
1500 linear feet and horizontal measurement on the Podunk
lode, vein, ledge, or deposit, along the vein thereof, with all its dips,
angles, and variations, as allowed by law, together with. . . .150 feet on
each side of the middle of said vein at the surface, so far as can be deter-
mined from present developments, and all veins, lodes, ledges, or deposits
and surface ground within the lines of said claim 140 feet running
N. 41° 14' W from centre of discovery cut and 1360....
feet running S. 41° 14' E from centre of discovery. . . .cut , said
discovery cut being situate upon said lode, vein, ledge, or deposit,
and within the lines of said claim in Pike's Peak Mining District,
County of El Paso , and State of Colorado, described by metes and
bounds as follows, to wit:
Beginning at Cor. No. i whence the SW. Cor. Sec. 17, T. 15 S.,
R. 69 W. bears S. 38° 2' W. 1465 ft., thence S. 41** H' E. 1500 ft. to Cor,
No. 2, thence S. 48° 46' W. 300 ft. to Cor. No. 3, thence N. 41 ** 14' W.
1500 ft. to Cor. No. 4, thence N. 48° 46' E. 300 ft. to Cor. No.. i, the place
of beginning This beinj? the same lode originally located on
the 6th day of April , 1888, and recorded on the 14th
day of June 1888, in Book 3 , Page 48 , in the office
of the Recorder of El Paso County. This further additional and
amended certificate of location is made without waiver of any previously
acquired rights, but for the purpose of correcting any errors in the orig-
inal location, description, or record, and of taking in and acquiring all
forfeited or abandoned overlapping ground, and of taking in any part of
any overlapping claim which has been abandoned, and of securing all the
benefits of said section 2409 of the General Statutes of Colorado.
Said lode was discovered the 1st
« dayof April , A.D. 18S8.
Attest :
Date of additional and amended certificate
June 14th , 1891.
.John H. Routt.
SBAU
SKAL.'
SBAL.'
SEAL.
SEAL.]
SEAL.'
LOCATION CERTIFICATE OF MILL SITE.
.To all whom these presents may concern:
Know ye. That I, A. E. Lowe, of the Coimty of Arapahoe, and State
of Colorado, do hereby declare and publish as a legal notice to all the
world, that I have a valid right to the occupation, possession, and enjoy-
ment of all and singular that tract or parcel of land, not exceeding five
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APPENDIX B.
673
acres, situate, lying and being in the County of El Paso, and State of
Colorado, bounded and described as follows:
POORMAN MILL SITE.
Beginning at Cor. No. i, whence the NE. Cor. Sec. 19, T. 14 S., R.
69 W. bears N. 46° 48' E. 3416.9 ft., thence S. 85° 50' W. 921.2 ft. to Cor.
No. 2, thence N. zi° 00' E. 467.66 ft. to Cor. No. 3, thence S. 64° 00' E.
700 ft. to Cor. No. I, the place of beginning.
Containing 4 acres, more or less.
Variation 14° 45' E.
Together with all and singular the hereditaments and appurtenances
thereto belonging or in any wise appertaining.
Witness my hand and seal this 5tli day of December, A.D. 1891.
[seal.] a. E. LOWE.
(4—694.)
SURVEY NO. 8000 A.
TITLE PAGE TO REPORT UNDER CIRCULAR " N " OF SEPT. 23, 1882.
REPORT
Under General Land Oflfice Circular " N " of September 23, 1882 upon
the Placer Mining Claim known as the Cumro placer , claimed
by T. E. Jenkins et al , situate in Pike's Peak Mining Dis-
trict , El Paso. .. .County Colorado , embracing. .. .32.07
acres, and forming a portion of the S. half of the SW. quarter. .. .in Sec.
17 , Town 14 S , Range 69 W. of the 6th P. M
Examination made.
..February 15th , 1893.
By A. L. Hawley ,
U. S. Deputy Mineral Surzfeyor.
SURVEY NO. 8000 A.
CUMRO PLACER.
The soil embraced in this claim consists of decomposed
mineral-bearing granite on the mountain slopes, and auriferous
sand and gravel along the creek bottom, all covered with a
thin layer of loam and alluvium supporting a scant growth of
grass and sage brush, with scattering pine, spruce, cedar, and
Cottonwood timber.
The only stream passing through this claim is Cumro creek,
4 ft. wide and about 2 ft. deep, which crosses the extreme
southeast corner.
A log cabin, the west corner of which bears from Cor. No.
13 S. 40° E. 120 ft., 12 X 16 ft., course of long sides N. 44° E.
The surface and underground workings on this claim con-
sist of:
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674
SU/^VEy/XG.
A tunnel, the mouth of which bears from Cor. No. 7 N. 67°
48' E. 582 ft., 5x6 ft., running N. 10° 44' W. 515 ft. to breast.
A shaft, which bears from Cor. No. 20 S. 48° 30' W. 305 ft.,
3 X 5 ft., 12 ft. deep in earth and rock.
Placer workings, the centre of the northeasterly end of which
bears from Cor. No. 15 N. 46° W. 285 ft., averaging 40 ft. wide
and 8 ft. deep, and extending S. 62^ W. 120 ft. along the bed
of Cumro creek.
The nearest post-office to the claim is Jamestown, a mining
camp of about 300 population, located on Brush creek, about 2
miles south of the claim. The nearest railroad station is Tie
Siding, a spur and flag station on the Denver, Apex and West-
ern R. R., at the conriuence of Cumro and Plum creeks, about
6 miles southwesterly from the claim.
Other than the system of lode deposits adjoining and form-
ing a part of this claim, there are none nearer than Carbonate,
situate about 4 miles to the northeast.
This claim is peculiarly adapted for placer-mining purposes,
inasmuch as the contour of the surface and the character and
nature cf the scil are such that it can be most advantageously
and cheaply worked by hydraulic giants and the tailings be
rapidly and easily disposed of. Cumro creek carries about
50 cu. ft. of water per second during the dry season, being an
abundance of water for working the c.aim. As yet no water
has been taken upon the claim for its development, except in
washing the placer workings hereinbefore described; but, by
a survey, it has been found that by a ditch not over one mile
in length water can be taken from Cumro creek onto the
highest portions of the claim. It being the express intention
of the claimants to work the claim in this manner.
The works and expenditures made by the claimants for the
development of the claim consist of the placer workings de-
scribed under paragraph c of this report.
There are no mines, salt licks, salt springs, or mill seats
upon this claim.
/
(4—695.)
OATH OF UNITED STATES DEPUTY MINERAL SURVEYOR.
UNDER GENERAL LAND OFFICE CIRCULAR " N " OF SEPT. 2^, 1882.
I, A. L. Hawley...., U. S. deputy mineral surveyor, do solemnly
swear that in pursuance of an order received from the U. S. Surveyor-
General for Colorado , dated .... February 6th , 1893, I have
made, under the provisions of General Land Office Circular '* N," ap-
proved September 23, 1882, a personal and thorough examination, upon
the premi:es, of the placer mining claim of T. E. Jenkins et al ,
known as the Cumro placer , situate in.... Pike's Peak.. . .Mining
District, El Paso County Colorado ,. embracing. .. .32.07
acres and forming a portion of the S. 1/2 of the SW. % of Sec. 17...., in
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APPENDIX B. 675
Township No. 14 S , Range No 69 W. of the 6th P. M , and
that my report of such examination, hereto attached, is specific and in de-
tail, and is a full and true statement of the facts upon all the points speci-
fied in said circular.
A. L. Hawley ,
U. S. Deputy Mineral Surveyor.
Subscribed and sworn to by the said A. L. Hawley , U. S.
deputy mineral surveyor, before me, a notary public in and for El
Paso County, Colorado , this 20th. . . .day of February, 1893.
[seal. J B. F. Clark
Notary Public
My commission expires December 20, 1893.
(4-696.)
CORROBORATIVE AFFIDAVIT UNDER GENERAL LAND
OFFICE CIRCULAR "N" OF SEPT. 23, 1882.
Stats df Colorado,)
County) of,, El Paso..,\ "*
....W. H. Wilson and J. P. Thompson , being first duly
swora, severally depose and say that he is personally and well acquainted
with the placer mining claim of T. E. Jenkins et at , known as
the Cumro placer , situate in Pikers Peak Mining District
, El Paso County, Colorado embracing 32.07 acres and
forming a portion of the S. J/2 of the SW. J4 oi Sec. 17, in Town-
ship No 14 S. Range No 69 W. of the 6tir P. M ; and
also with the character of all the land included in said claim, and has
been so acquainted for 10 and 12 years last past; that his knowl-
edge of said claim and land is derived from prospecting the ground
and working the claim and is such as to enable him to testify under-
standingly with regard thereto; that he has carefully read the foregoing
report of A. L. Hawley U. S. deputy mineral surveyor, and that
to his own personal knowledge said report is in all respects true and
accurate.
W. H. Wilson
J. P. Thompson
Subscribed and sworn to by the above-named persons this 20th
day of February , 1893.
[seal.] B. F. Clark ,
Notary Public
My commission expires December 20, 1893.
(4-687.)
SURVEYOR-GENERAL'S CERTIFICATE OF APPROVAL OF
FIELD NOTES AND SURVEY OF MINING CLAIM.
Department of the Interior,
Office of U. S. Surveyor-General,
-18—.
I, U. S. Surveyor- General for , do hereby certify that the fore-
going and hereto attached field notes and return of the survey of the
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676 SURVEYING,
mining claim of , known as the , situate in
mining district, County, , in Section , Township No.
, Range No. , designated as Survey No. , executed by
, U. S. deputy mineral surveyor, , 18 , under my instruc-
tions dated , 18 , have been critically examined and the neces-
sary corrections and explanations made, and the said field notes and re-
turn, and the survey they describe, are hereby approved. A true copy
of the copy of the location certificate filed by the applicant for survey is
included in the field notes.
JJ. S, Surveyor-General
For
(4-688.)
UNITED STATES SURVEYOR-GENERAL'S FINAL CERTIFI-
GATE ON FIELD NOTES.
Department of the Interior,
Office of U. S. Surveyor-General,
-i&~.
I, U. S. Surveyor-General for , do hereby certify that the fore-
going transcript of the field notes, return, and approval of the survey of
the mining claim of , known as the , situate in mining
district, County, , in Section , Township No. ,
Range No. , and designated as Survey No. , has been cor-
rectly copied from the originals on file in this office; that said field notes
furnish such an accurate description of said mining claim as will, if in-
corporated into a patent, serve fully to identify the premises, and that
such reference is made therein to natural objects or permanent monu-
ments as will perpetuate and fix the locus thereof.
And I further certify that five hundred dollars' worth of labor has been
expended or improvements made upcn said mining claim by claimant
or gjantors, and that said improvements consist of , and
that no portion of said labor or improvements has been included in the
estimate of expenditures upon any other claim.
I further certify that the plat hereof, filed in the U. S. Land Office
at , is correct and in conformity with the foregoing field notes.
U. S. Surveyor-General^
For
APPENDIX A,
CIRCULAR TO APPLICANTS.
To Applicanis for Mineral Survey Orders:
You will observe the following requirements in the conduct of your
business with the Surveyor-General's Ofiice, the same being based upon
the United States mining laws and circular and special instructions from
the Commissioner of the General Land Office:
I. All applications for survey orders, descriptive reports on placer
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APPENDIX B. ^77
claims, or certificates of five hundred dollars' expenditure, should be ad-
dressed to the Surveyor-General * and be signed by the claimants, their
agent or attorney.
2. Each application should contain:
(a) The name of the claimant iri full, and as it is desired to appear in
the application for patent.
(6) The name of each location embraced in the claim.
(c) The name of the land and mining districts in which the claim is
located.
(rf) The name of the United States deputy mineral surveyor to whom
it is desired the order shall be issued.
(For form of application see page 655.)
3. You are required to file with each application for survey order, a
copy of the record of location of the claim, properly certified by the re-
corder of the county or mining district where the claim is situate.
4. The deputy mineral sur\eyor is required to survey the claim in
strict conformity with or within the lines of the location upon which the
order of survey is based. You are, therefore, advised before filing your
application to see that your location has been made in compliance with
the law and regulations, and that it properly describes the_ claim for which
the patent is sought.
The act of Congress of May 10, 1872, expressly provides that " the
location must be distinctly marked on the ground, so that its boundaries
can be readily traced," and " that all records of mining claims hereafter
made shall contain the name or names of the locators, the date of loca-
tion, and such a description of the claim or claims, located by reference
to some natural object or permanent monument, as will identify the
claim."
** These provisions of the law must be strictly complied with in each
case to entitle a claimant to a survey and patent, and therefore should
a claimant under a location made subsequent to the passage of the mining
act of May 10, 1872, who has not complied with said requirements in re-
gard to marking the location upon the ground, and recording the same,
apply for a survey, you will decline to make it."
" The only relief for a party under such circumstances, will be to make
a new location in conformity to law and regulations, as no case will be
approved by this office, unless these and all other provisions of law are
substantially complied with." (See General Land Office circular dated
Nov. 20, 1873.)
5. Par. 99, General Land Office circular, of Dec. 10, 1891, edition Dec.
I, 1894, relating to the expense of office work connected with the survey
of mineral claims, reads as follows:
'* With regard to the platting of the claim and other olHce work in the
Surveyor-General's office, that officer will make an estimate of the cost
thereof, which amount the claimant will deposit with any assistant United
States treasurer, or designated depository , in favor of the United States
treasurer, to be passed to the credit of the fund created by * individual
depositors for surveys of the public lands,' and file with the Surveyor-
General duplicate certif.cates of such deposits in the i sual manner."
6. The various Surveyors-General have adopted schedules of rates for
* See pa^e 682 for list of oflQces of U. S. Sunreyors-Genaral.
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678
SURVEYING.
office work, and an estimate of the cost in any particular case may be
had upon application.
Should an applicant deem an estimate excessive, he will be allowed
the right of appeal to the General Land Office in the usual manner.
In transmitting such an appeal the Surveyor-General should transmit
therewith a full report.
7. Should the office work in any case amount to more than the esti-
mate, or if an amended order is issued, an additional deposit will be re-
quired.
8. In districts where there are no United States depositories, you
should deposit with the nearest assistant United States treasurer, or de-
pository, and in all cases immediately forward the original certificate to
the Secretary of the Treasury and the duplicate to the Surveyor-Generars
office, retaining the triplicate for your own use and security. Under no
circumstances will the deposit be made by the Surveyor-General. (See
paragraph 5, preceding.)
9. An application for an amended survey order must be accompanied
with a statement setting forth fully the reasons for the proposed amend-
ment and all the material facts in the matter.
10. If, after having obtained a survey order, you should abandon your
purpose of having a survey made, you can apply the deposit, less the
amount estimated for office expenses already incurred, on a new survey
if one is desired.
11. Upon discovery of any error or defect in an order you are re-
quested to return it to the Surveyor-Generars Office for correction or
amendment.
12. If, after having obtained an order for survey, you should find that
the record of location does not practically describe the location as staked
upon the ground, you should file a certified copy of an amended location
certificate, correctly describing the claim, and obtain an amended order
for survey. If a relocation of tie claim is made embracing ground not
included in the oripjinal order, or other material change is made, you will
abandon the original number of the order for survey, and a new order
will be issued in. which a number in the current series will be substituted.
13. The order of approval of surveys of mineral claims is prescribed
by General Land Office circular dated March 3, 1881, as follows:
*' The mining survey first applied for shall have the priority of action
in all its stages in the office of the Surveyor-General, including the de-
livery thereof, over any other survey of the same ground or any portion
thereof.
** The Surveyor-General should not order or authorize a survey of a
claim which conflicts wi;h one previously applied for until the survey first
applied for has been completed, examined, approved, and platted, and the
plats delivered.
" When the conflict does not appear uptil the field notes of the re-
spective surveys are returned, then the survey first applied for should be
first examined, approved, and platted, and the plats delivered before the
field notes of the survey last applied for are taken up for examination or
plats constructed.
" When the survey first authorized is not returned within a reasonable
period, and the applicant for a conflicting survey makes affidavit that he
believes (stating the reasons for his belief) that such first applicant has
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APPENDIX B. 679
abandoned his purpose of having a survey made, or is deferring it for
vexatious purposes, to wit, to postpone the subsequent applicant, the
Surveyor-General shall give notice of such charges to such first applicant,
and call upon him for an explanation under oath of the delay. He shall
also require the deputy mineral surveyor to make a full statement in
writing, explanatory of the delay; and if the Sur yor-General shall con-
clude that pood and rufficieat reasons for such delay do not exist, he shall
authorize the applicant for the conflicting survey to proceed with the
same; otherwise the order of proceedings shall not be changed.
** Whenever an applicant for a survey shall have reason to suppose that
a conflicting claimant will also apply for a survey for patent, he may
give a notice in writing to the Surveyor-General particuhrly describing
such conflicting claim, and file a copy of the notice of location of such
conflicting claim. In such case the Surveyor-General will not order or
authorize any survey of such conflicting claim until the survey first applied
for has been examined, completed, approved, and platted, and the plats
delivered."
14. You have the option of employing any United States deputy min-
eral surveyor in the district to execute the order of survey, and must
make satisfactory arrangements with such surveyor for the payment for
his services and those of his assistants in making the survey, as the
United States will not be held responsible for the payment of the same.
The duty of the deputy surveyor in any particular case ceases when he
has executed the survey and returned the same to this office. He is not
allowed to prepare for the mining claimant the papers in support of an
application for patent, being precluded from acting either directly or in-
directly as attorney in mineral claims. (Sec. 2334, United States Revised
Statutes; see Appendix B hereof.)
15. You are advised of your right to appeal to the Commissioner of
the General Land Office from the approval or disapproval of the survey of
your claim. The appeal must be in writing or in print, should set forth
in brief and clear terms the specific points of exception to the ruling ap-
pealed from, and should be transmitted through the Surveyor-General's
Office.
APPENDIX B,
Sections of the U. S. Revised Statutes and Paragraphs of the
Mining Circular of December id, 1891, relative to Surveys of
Mining Claims.
Sec. 2325. A patent for any land claimed and located for valuable
deposits may be obtained in the following manner: Any person, associa-
tion, or corporation authorized to locate a claim under this chapter,
having claimed and located a piece of land for such purposes, who has,
or have, complied with the terms of this chapter, may file in the proper
land-ofYice an application for a patent, under oath, showing such com-
pliance, together with a plat and field notes of the claim or claims in
common, made by or under the direction of the United States Surveyor-
General, showing accurately the boundaries of the claim or claims, which
shall be distinctly marked by monuments on the ground. ♦ ♦ ♦
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68o SURVEYING.
Par. 28. The claimant is required in the first place to have a correct
survey of his claim made under authority of the Surveyor-General of the
State or Territory in which the claim lies; such survey to show with
accuracy the exterior surface boundaries of the claim, which boundaries
are required to be distinctly marked by monuments on the ground. Four
plats and one copy of the original field notes, in each case, will be pre-
pared by the Surveyor-General; one plat and the original field notes
to be retained in the office of the Surveyor-General, one copy of the plat
to be given the claimant for posting upon the claim, one plat and a
copy of the field notes to be given the claimant for filing with the proper
register, to be finally transmitted by that officer, with other papers in
the case, to this office, and one plat to be sent by the Surveyor-General
to the register of the proper land district to be retained on his files for
future reference. As there is no resident Surveyor-General for the State
of Arkansas, applications for the survey of mineral claims in said State
should be made to the Commissioner of this office, who, under the law,
is ex oMcio the U. S. Surveyor-General.
45. The Surveyors-General should designate all surveyed mineral
claims by a progressive scries of numbers, beginning with survey No.
^y, irrespective as to whether they are situated on surveyed or unsurveyed
lanoj, the claim to be so designated at date of issuing the order therefor,
in addition to the local designation of the claim; it being required in all
cases that the plat and field notes of the survey of a claim must, in addi-
tion to the reference to permanent objects in the neighborhood, describe
the locus of the claim, with reference to the lines of public surveys, by
a line connecting a corner of the claim with the nearest public corner of
the United States surveys, unless such claim be on unsurveyed lands at
a distance of more than two miles from such public corner, in which
latter case it should be connected with a United States mineral monu-
ment. Such connecting line must not be more than tivo miles in length
and should be measured on the ground direct between the points, or
calculated from actually surveyed traverse lines if the nature of the country
should not permit direct measurement. If a regularly established survey
corner is within two miles of a claim situated on unsurveyed lands, the
connection should be made with such corner in preference to a con-
nection with a United States mineral monument. The connecting line
must be surveyed by the deputy mineral surveyor at the time of his
making the particular survey, and be made a part thereof.
46. Upon the approval of the survey of a mining claim made upon
surveyed lands, the Surveyor-General will prepare and transmit to the
local land office and to this office a diagram tracing showing the portions
of legal 40-acre subdivisions made fractional by reason of the mineral sur-
vey, designating each of such portions by the proper lot number, be-
ginning with No. I in each section and giving the area of each lot.
47. The survey and plat of mineral claims, required by section 2325,
Revised Statutes of the United States, to be filed in the proper land office,
with application for patent, must be made subsequent to the recording of
the location of the mine; and when the original location is made by sur-
vey of a United States deputy surveyor, such location survey cannot
be substituted for that required by the statute, as above indicated.
48. The Surveyor-General should derive his information upon which
to base his certificate as to the value of labor expended or improvements
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APPENDIX B. 68 1
made from his deputy who makes the actual survey and examination upon
the premises, and such deputy should specify with particularity and full
detail the character and extent of such improvements.
49. The following particulars should be observed in the survey of every
mining claim:
(i) The exterior boundaries of the claim should be represented on
the plat of survey and in the field notes.
(2) The intersection of the lines of the survey with the lines of con-
flicting prior surveys should be noted in the field notes and represented
upon the plat.
(3) Conflicts with unsurveyed claims, where, the applicant for survey
does not claim the area in conflict, should be shown by actual survey.
(4) The total area of the claim embraced by the exterior boundaries
should be stated, and also the area in conflict with each intersecting sur-
vey, substantially as follows:
Acres.
Total area of claim 10.50
Area in conflict with survey No. 302 1.56
Area in conflict with survey No. 0^8 2.33
Area in conflict with Mountain Maid l)de mininf; claim, unsurveyed 1.48
It does not follow that because mining surveys are required to exhibit
all conflicts with prior surveys the areas of conflict are to be excluded.
The field notes and plat are made a part of the application for patent,
and care should be taken that the description does not inadvertently ex-
clude portions intended to be retained. It is better that the application
for patent should state the portions to be excluded in express terms. A
survey executed as in the example given will enable the applicant for
patent to exclude such conflicts as may seem desirable. For instance,
the conflict with survey No. 302 and with the Mountain Maid lode claim
might be excluded and that with survey No. 948 included.
(For paragraphs 50 and 51, see page 653.)
Sec. 2327. The description of vein or lode claims, upon surveyed lands,
shall designate the location of the claim with leference to the lines of
the public surveys, but need not conform therewith; but where a patent
shall be issued for claims upon unsurveyed lands, the surveyor-general,
in extending the surveys, shall adjust the same to the boundaries of such
patented claim, according to the plat or description thereof, but so as in
no case to interfere with or change the location of any such patented
claim. (See paragraph 46, page 652.)
Sec. 2331. Where placer claims are upon surveyed lands, and conform
to legal subdivisions, no further survey or plat shall be required, and all
placer-mining claims located after the tenth day of May, eighteen hundred
and seventy-two, shall conform as near as practicable with the United
States system of public-land surveys, and the rectangular subdivisions of
such surveys, and no such location shall inc'ude more than twenty acres
for each individual claimant; but where placer claims cannot be con-
formed to legal subdivisions, survey and plat shall be made as on un-
surveyed lands; and where by the segregation of mineral lands in any
legal subdivision a quantity of agricultural land less than forty acres re-
mains, such fractional portion of agricultural land may be entered by any
party qualified by law, for homestead or pre-emption purposes.
Sec. 2334. The Surveyor-General of the United States may appoint
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6^2 surveViM,
in each land district containing mineral lands as many competent sur-
veyors as shall apply for appointment to survey mining claims. The ex-
penses of the survey of vein or lode claims, and the survey and subdivision
of placer claims into smaller quantities than one hundred and sixty acres,
together with the cost of publication of notices, shall be paid by the appli-
cants, and they shall be at liberty to obtain the same at the most rea-
sonable rates, and they shall also be at liberty to employ any United
States deputy surveyor to make the survey. The Commissioner of the
General Land Office shall also have power to establish the maximum
charges for surveys. ♦ ♦ ♦
98. The Surveyors-General of the several districts will, in pursuance
of said law, appoint in each land district as many competent deputies for
the survey of mining claims as may seek such appointment: it being
distinctly understood that all expenses of these notices and surveys are
to be borne by the mining claimants and not by the United States; the
system of making deposits for mineral surveys, as required by previous
instructions, being hereby revoked as regards Md work; the claimant
having the option of employing any deputy surveyor within such district
to do his work in the field.
99. With regard to the platting of the claim and other oMce work in
the Surveyor-General's Office, that officer will make an estimate of the
cost thereof, which amount the claimant will deposit with any assistant
United States treasurer or designated depository in favor of the United
States Treasurer, to be passed to the credit of the fund created by " in-
dividual depositors for surveys of the public lands," and file with the
Surveyor-General duplicate certificates of such deposft in the usual man-
ner.
100. The Surveyors-General will endeavor to appoint mineral deputy
surveyors, so that one or more may be located in each mining district for
the greater convenience of miners.
loi. The usual oaths will be required of these deputies and their assist-
ants as to the correctness of each survey executed by them.
The duty of the deputy mineral surveyor ceases when he has executed
the survey and returned the field notes and preliminary plat thereof with
his report to the Surveyor-General. He will not be allowed to prepare
for the mining claimant the papers in support of an application for patent,
or otherwise perform the duties of an attorney before the land omce in
connection with a mining claim.
The Surveyors-General and local land officers are expected to report
any infringement of this regulation to this office.
DESCRIPTIVE REPORTS ON PLACERS.
Par. 6^, Mining Cik ular.
(2) Section 2395. Revised Statutes (subdivision 7), requires the sur-
veyor to '* note in his field books the true situation of all mines, salt licks,
salt springs, and mill seats which come to his knowledge;" also "all
watercourses over which the lines he runs may pass." It further requires
him to " note the quality of the lands." These descriptive notes are re-
quired by subdivision 8 to be incorporated in the plat by the Surveyor-
General.
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APPENDIX B. 683
(3) If these duties have been performed, the public surveys will fur-
nish a reasonable guide to the district officers and to claimants in prose-
cuting their applications. But experience has shown that great neglect
has resulted from inattention to the law in this respect, and the regular
plats are of very little value in the matter. It will, therefore, be required
in the future that deputy surveyors shall, at the expense of the parties,
make full examination of all placer claims surveyed by them, and duly
note the facts as specified in the law, stating the quality and composition
of the soil, the kind and amount of timber and other vegetation, the locus
and size of streams, and such other matters as may appear upon the sur-
face of the claim. This examination should include the character and ex-
tent of all surface and underground workings, whether placer or lode, for
mining purposes.
(4) In addition to these data, which the law requires to be shown in
all cases, the deputy should report with reference to the proximity of
centres of trade or residence; also of well-known systems of lode deposit
or of individual lodes. He should also report as to the use or adaptability
of the claim for placer mining; whether water has been brought upon it
in sufficient quantity to mine the same, or whether it can be procured for
that purpose; and, finally, what works or expenditures have been made
by the claimant or his grantors for the development of the claim, and their
situation and location with respect to the same as applied for.
(5) This examination should be reported by the deputy under oath
to the Surveyor-General, and duly corroborated; and a copy of the same
should be furnished with the application for patent to the claim, consti-
tuting a part thereof, and included in the oath of the applicant.
CONTESTS BETWEEN MINERAL AND AGRICULTURAL
CLAIMANTS.
Segregation Surveys.
114. When the case comes before this office [General Land Office],
such decision will be made as the law and the facts may justify; and in
cases where a survey is necessary to set apart the mineral from the agricul-
tural land, the necessary instructions will be given to enable the proper
party, at his own expense, to have the work done, at his option, either by
United States deputy, county, or other local surveyor; the survey in
such case, where the claims to be segregated are vein or lode claims, must
be executed in such manner as will conform to the requirements in section
2320, U. S. Revised Statutes, as to length and width and parallel end
lines.
115. Such survey when executed must be properly sworn to by the
surveyor, either before a notary public, officer of a court of record, or be-
fore the register or receiver, the deponent's character and credibility to be
properly certified to by the officer administering the oath.
116. Upon the filing of the plat and field notes of such survey, duly
sworn to as aforesaid, you [Register and Receiver] will transmit the same
to the Surveyor-General for his verification and approval; who, if he
finds the work correctly performed, will properly mark out the same upon
the original township plat in his office, and furnish authenticated copies
of such plat and description both to the proper local land office and to this
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684 SURVEYING.
office, to be affixed to the duplicate and triplicate township plats re-
spectively.
117. With a copy of plat and description furnished the local office
and this office, must be a diagram tracing, verified by the Surveyor-Gen-
eral, showing the claim or clamis segregated, and designating the separate
fractional agricultural tracts in each 40-acre legal subdivision by the
proper lot number, beginning with No. i in each section, and giving the
area in each lot, the same as provided in paragraph 45, in the survey of
mining claims on surveyed lands.
SURVEY OF ADVERSE CLAIMS.
Sec. 2326. Where an adverse claim is filed during the period of pub-
lication, it shall be upon oath of the person or persons making the
same, and shall show the nature, boundaries, and extent of such adverse
claim. * • *
86, In order that the '* boundaries " and " extent " of the claim may be
shown, it will be incumbent upon the adverse claimant to file a plat show-
ing his entire claim, its relative situation or position with the one against
which he claims, and the extent of the conflict. This plat must be made
from an actual survey by a United States deputy surveyor, who will
officially certify thereon to its correctness; and in addition there must
be attached to such plat of survey a certificate or sworn statement by the
surveyor as to the approximate value of the labor performed or improve-
ments made upon the claim by the adverse party or his predecessors in
interest, and the plat must indicate the position of any shafts, tunnels, or
other improvements, if any such exist, upon the claim of the party op-
posing the application, and by which party said improvements were made:
Provided, hotvever, That, if the application for patent describes the claim
by legal subdivisions, the adverse claimant, if also claiming by legal sub-
divisions, may describe his adverse claim in the same manner without
further survey or plat.
MINING DISTRICTS AND SURVEYORS-GENERAL.
Alaska, Sitka Williams, Louis L.
Arizona, Tucson Manning, Levi H.
California, San Francisco Green. William S.
Colorado, Denver Robinson, Thomas D.
Idaho, Boise City Straughan, Joseph C.
Montana, Helena Neill; John S. M.
Nevada, Reno Belknap, Clayton.
New Mexico, Santa Fe Easley, Charles F.
Oregon. Portland Arnold, John C.
South Dakota. Huron » Hughes, Richard B.
Utah. Salt Lake City Snow, George W.
Washington, Olmypia Watson, William P.
Wyoming, Cheyenne Thompson, John C.
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APPENDIX C
FINITE DIFFERENCES.
THE CONSTRUCTION OF TABLES.
In the accompanying figure the ordinates are spaced at the uniform
distance /apart. Let the successive values of these ordinates, and theii ^
several orders of differences, be represented by the following notation :
I I I i
Fig. 15a.
Values of the function, Ao, h\, ht, h^, A4, h%, h^
First order of differences, Ak^, ^'h^, ^'a^. ^'a^. ^'h^, ^'h^
Second '• ** J">lo. ^"Ap A"a^. ^"a„ ^"a,.
Third " '* A"a,, J'"a,, ^'"a,. J'"Ar
Fourth *• *• ^«'Ao, -^'^A,, ^^''Ar
etc., etc.
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686 SURVEYING.
We may now write
//, = A, + J' A, = Ao + ^'Ao + ^'Ao + ^"Ao = -*• + 2^'*o + -^">lo;
A, = ^, + J'a, = Ao + 3-^'a« + 3^" Ac + ^'"AoI
A,=:A, + 4^' A. + t^"A, + 4^"'A. + ^«'Ao;
» • >*» I « (« — i) .i/f I « (« — i) {n — 2) .,„ ,
hn = A. + nJ'A. + -777-^ ^0 H ^^-3 ^ ^« + «tc.
(I)
It is to be observed that the coefficients follow the law of the bino-
mial development. It is also seen that Xht first of the successive orders
of differences are alone sufficient to enable any term of the function to
be computed. We will now proceed to find these first terms of the
several orders of differences for any given equation.
Almost all functions of a single variable can be developed by the aid
of Maclaurin's Formula, in the form
^0 = Co + Cjto + C,jro» + r,x«» + C4Jf«* + etc (2)
If X take an increment ^j-. thus becoming Xu the change in jo will
be represented by A'y^ and its value will be the new value of the function
minus its initial value, or J'y^ =j/i — j'o. By putting jt+^j- for x in the
above equation, developing, subtracting the original equation, and re-
ducing, we would obtain
yx— y„= '/I'yc = (C, + 2C94ro + 3^»Jfo' + aCaXc*)^x
+ (a + 3C'.^o + 6C4X.Vx+«^« + 4^4X.)J«x+C»^*x, . (3)
assuming that the function stops with G-n*.
If Xx should now take another increment -^ jr. equal to the previous
one, we would have Xi = Xx -{- A^ and ^a = ji + ^y^. Now A' y is the
value A'y when jto has become jTi. and the /^>^<rr^;/r^ bet ween A'y^ and
A'y^ is the change in the value of A'y^ due to this change in x.
Hence A'y^ — A'y^ = A"y^,
To find the value of A"y^^ substitute x '\' Ax iox x in equation (3),
develop, subtract equation (3), reduce, and obtain
A"y^ = (2C, + 6r,xo + I2C4^.V*^ + (6C, + 24C4a:oM»-r + 14^^*0-. (4)
Similarly we find
A"'y^^[!bC^-\^^^C^,)A^x-\^^tC^A^x, (5)
2/»Vo = 24CiA*x (a constant) (6)
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APPENDIX C.
687
From tlie above development we see —
1. That the number of orders of differences is equal to the highest ex-
ponent of the variable imwlved, ihe last difference being a constant.
2. That if any initial value (^0) of the variable be taken, the first of
the several orders of differences can be obtained in terms of this initial
value, its constant increment, and the constant coefficients. This fur-
nishes a ready means of computing a table of values of the function, if
it can be represented in the form of equation (i). Evidently if the ini-
tial value of the variable (jr©) be taken as zero, the evaluation for the
several iniiial differences is much simplified, for then all the terms in .r
disappear. If the constant increment be also taken as unity, tiie labor
is still further reduced.
Example. — Construct a table of values of the function
>' = 50 — 400- + 20jr* + 4^ ■" ^*'
.... (7)
Let the initial value of the variable be zero and the increments unity.
Evaluating the initial differences by equations (3) to (6), we find, for jto = o,
and Jx =• I,
;-• = + 50;
^'jro» = C, -f- G + G + r« = - 17;
A'y^ = 2C, + 6C, + 14C4 = + 50;
^"y = 6Cs + 36C4 = — 12;
^•▼^0 = 24C4 = — 24.
From these initial values we may readily construct the following
tiM)]c:
Vah OS of
Values of
ist Diflfeienccs.
ad Differences.
3d Differences.
1
4th Differences.
A.
y*
^'r
A-,.
A-,.
A«V
0
50
— 17
I
33
+ 33
+ 50
— 12
2
66
+ 71
+ 38
- 36
- 24
3
^37
+ 73
+ 2
- 60
- 24
4
210
+ 15
- 58
- 84
- 24
5
225
— 127
- 142
— 108
- 24
6
98
- 377
— 250
- 132
- 24
7
- 279
— 759
- 382
etc.
etc.
8
— 1038
etc.
etc.
etc.
etc.
♦ Fig. 152 is the locus of this curve, the ordinates being taken from tbJs
column.
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688 SU/^VEVIKC.
The initial values in all the columns being given, the table is made
by continual additions, one column after another, working from right to
left. Thus, the 4th difference being constant, the initial value, —24, is
simply repeated indefinitely. The column of 3d differences is now com-
puted by adding continuously —24 to the preceding value. The column
of 2d differences is next made out, the quantity to be added each time
being the intervening 3d difference, which is not constant. In a similar
manner proceed with the column of ist differences, and finally with the
values of the function itself.
The above formulae apply to all functions of a single variable not
higher than the fourth degree. Evidently any of the C coefficients may
be zero, and so cause one or more of the powers of ;rto entirely disappear.
If the variable is involved to a higher degree than the fourth, a new de-
velopment may be made, or the initial values of the successive orders of
differences may be determined by simply evaluating the function for a
series of successive values of the variable, one more in number than the
degree of the equation, and then working out the successive columns of
differences from these until the last, or constant, difference is found.
The table may then be continued by combining these differences, as be-
fore. Thus in the above example the first five values of^ might have
been found by direct evaluation of the function for the corresponding
values of x, and then the successive differences taken out until the con-
stant fourth difference. — 24, was found. This can always be done with-
out resorting to any algebraic discussion as given above.
THE EVALUATION OF IRREGULAR AREAS.
The ordinates to any curve, as that in Fig. 152 for instance, may be
represented by such an equation as the last of equations, (i). where the
length of any ordinate is given in terms of its number from the initial or-
dinate, the value of this first ordinate, and the first of the successive
orders of differences. This equation is
hn = ho + n/l'h^ + -777-^ >fc. + fTT;-^ -^ h. + etc.,
where hn is the «th, and therefore any ordinate to the curve. The con-
stant distance between the ordinates apparently does not enter the equa-
tion, but it is really represented in the several ^'s.
By the calculus the area of any figure included between any curve,
the axis of abscissas, and two extreme ordinates \s A = I kdx, where h
<
is the general value of an ordinate, = hn in the above equation, where it
is shown to be a function of «. Also x = nl where / is the constant
distance between ordinates, whence dx = Idn, Substituting these val-
ues of h and dx, we have
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APPENDIX C\ 6j9
A - C' kdx = C" hnidn = l P" hndn^A h^ T" dn'\-^'M^ r"" ndn
+ Tj-*y"'''(«-')(^-2)(^-3)^« + etc."|. . . . (8)
Integrating this equation, we obtain
' ^5040 288 730 64 1080 18 ^^
From the schedule of differences on p. 605 we may at once find the
initial values of the several orders of differences in terms of the succes-
sive values of the function. Thus
^"a, = ^A, — ^'h^ = ^j — 2/5, + h%\
J "a, = ^"h^ - ^"k^ - A'k^ — 2A'k^ + A'h^ = At " 3^, + 3^, - ^,;
= >44 - 4^t + 6A, — 4^1 + ^..
Again, the coefficients follow the law of the binomial development,
and we may write
^ , , t ^(f*—^), «(«— 1)(« — 2), , , ^
J«Ao = Ai, - »/4i. - I + '^ ^ ^>» -a ""T2 i " * - 3 + ^^^ • ^'°^
By the aid of this equation we may now substitute for the several
initial differences in equation (9) liieir values in terms of the successive
values of the function. Also for any area divided into n sections by or-
dinates, uniformly spaced a distance / apart, equation (9) will give the
area in lerms of /, «, and the several ordinates, when these latter are sub-
stituted for the J*s by means of eq. (10).
Thus, for « = I, equation (9) becomes
A = /(ho+^J'Ao) = \{/^o + Ai), (II)
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690 SURVEYING.
which is I lie Trapezoidal Rule,
For // = 2,
A =/(2>io + 2J'Ao + (| - iM">k.) = /(iM-f>*i+i'*.) = ^(>4.+ 4^i+>^t)., (")
which IS called Simpson's lr h tile.
If / = 2/ = total length of figure, this formula becomes
^=^(>4. + 4>4i + ^«) (la^
which is the well-known /orw of the Prismoidal Formula, and it would
be that formula if areas were substituted for ordinates.
^ = |-^>*. + 3>4. + 3>4. + >iA (13)
which is called Simpson's t Rule.
^ = t1-[7(>4. + >^4) + 32(>4i + >».)+I2>IJ (14)
45
If « = 6,
If now the coefficient of J'^Uo be changed from -,*|V to -j^^, which
would not affect curves of a degree less than the sixth, the resulting
equation, when the ^'s are substituted for the -^'s, takes the following
very simple form :
^=A^[>io + >i, + >54 + >5. + 5(>5. +>». + >».) + >*.]. . . (15)
which is called IVeddeVs Rule.
For a greater number of ordinates than seven, it is best to use either
equation (12). (13), or (15) several times, as the formulae become verj
complicated for « > 6.
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APPENDIX D.
DERIVATION OF FORMULAE FOR COMPUTING GEOGRAPH-
ICAL COORDINATES AND FOR THE fRO-
JECTION OF MAPS.*
Let Fig. 1 53 represent a distorted meridian section of the earth.
Let a = the major and b the minor semi-axes.
a " b
Theo e = = the elhpticity.
The eccentricity Is given by
6* s= 5—, whence i — /• =s -5.
The line nm = TV is the normal to the curve at #» ;
the angle tied = A is the geocentiic latitude ;
while nld = Z is the geodetic latitude.
The geodetic latitude is always understood, as it is the latitude ob-
tained from astronomical observations.
It is desirable to find the length of the line «/, of the normal nm, and
of the radius of curvature /V, all in terms of e, L, and a. Also to find
the geocentric latitude in terms of a» b, and Z.
To find nl, we have
For the ellinse.
dj^
dy b]x^
dx "" ay
Whence «/= 4/y + $:^ = V^^' + <' -'^-••
(I)
(a)
• See Chapters XIV. and XV. for the use of the formulae.
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692 SURVEYING.
But the equation of the ellipse in terms of its eccentricity is
whence «/= V;/»<f» + «« (i - ^. • (S
Fig. 153.
Squaring, remembering that^' = «/sin L, we have, after reducing,
«^-(,«^sin»ZH *^
To find the length of the normal nm = N, we have
— dy 3*
But i£?'=iK/ian<//f/=^~=-jj: = (i — /•)*; ^4)
whence ^^ = ^^ = i^ = (^ ,/,;„, z)^- ...... (B)
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APPENDIX /). 693
To find A, the geocentric latitude in terms of a, b, and Z, we have
X = ncd ; L = nld.
Since both have the common ordinate nd, we may write
tan X : tan Lw dl \ dc.
Bat dl^-zx from (4), and dctsx^
jrhence tan A = -, tan Z •••••• (Q
To find the radius of curvature, R, we have, in general,
(.+£)'
</^
For the ellipse. ±^ - Jty ««<« ^ = -i5i»
;e = ^Sf±^ <Q
To get this in terms of a, e, and Z, we have, from Fig. 153,
f ^nl sin» Z = — ^^ _L____.
I — ^ sin« Z
;
Also from the equation of the ellipse in terms of its eccentricity we
have
^^^. y ^a»(i-sin>Z)
I— If* I— ^sin*Z'
We may now find
■^ ' I — r* sin» Z
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694
SURVEYING,
(^y + ^:r»)} =
(^b^
(I - ^ sin* Z)i'
(n
Substituting this in (6), we obtain
ip=i'.
g(l-^
a • (I - ^« sin« Z)J " (i - ^ sin« L)X
(D)
The radius of curvature of the meridian, R, and the radius of curva-
ture of the ^eat circle perpendicular to a given meridian at the point
where they mtersect, which is the normal, A^, are the most important
functions in geodetic formulae. We will now derive the equations used
Fia X54.
on the U. S. Coast and Geodetic Survey for computing geodetic positions
from the results of a primary triangulation.
In Fig. 154, let A and B be two points on the surface of the earth,
which were used as adjacent trianfjulation-stations. The distance oetwecn
them, the azimuth of the line AB at one of the stations, and the latitude
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APPENDIX D. 695
and longitude of one station are supposed to be known ; the latitude and
longitude of the other station, and the back azimuth of the line joining
them, are to be found.
Let L' = known latitude oi B\
L = unknown latitude of A ;
K = known length of line AB reduced to sea-level;
s = length of arc AB = jrr;
Z = known azimuth of BA at B\
Z =: unknown azimuth of AB at A\
M' •= known longitude of B\
M := unknown longitude of A,
The angle APB formed by the two meridional planes through A and
B is the difference of longitude J/— Af = AM,
The difference of latitude is, L — L' ^ AL = B/ in the figure. A/ is
the trace of a parallel of latitude througli A and / is its intersection with
the meridian through B. AP' is the trace of a great circle through A
perpendicular to the meridian through B, and P* is the point of its inter-
section with tint meridian.
The normals are Bn' = iV* and An = iV. The radii of curvatvre are
Br' = R and Ar - K.
The latitude and longitude of A, and the azimuth of the line AB from
A towards B, can now be found by solving the spherical triangle APB,
Thus Z = 90** - ^Z'; Af ^ M' ^ Al\ and Z = 180** - PAB,
Although the line AB lies on the surface of a spheroid, if a sphere be
conceived such that its surface is tangent internally to the surface of the
spheroid on the parallel of latitude passing through the middle point of
the line AB, then this line will lie so nearly in the surface of the sphere,
that no appreciable error is made by assuming it to be in its surface. The
triangle ABP then becomes a triangle on the surface of the tangent
sphere, and hence is a true spherical triangle. The sphere is defined by
taking its radius equal to the normal to the meridian at the mean lati*
tude ot the points A and B, Since this mean latitude is unknown, the
formulae are first derived for the latitude of B, L\ and then a correction
applied to reduce it to the mean latitude.
THE DIFFERENCE OF LATITUDE.
Let it first be required to find L from L\ or find AL =L — L\
If we write /, A for the co-fatitudes of Z, L\ and ^ forZ' — 180*, we .
have, from the spherical triangle ABP,
cos / = C09 / cos ^ 4~ sill ^ ^^^ ^ ^^^ ^* • • • • • (8}
41
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696
SURVEYING.
By means of Taylor's Formula we may find the value of / in ascending
powers of s^ and since s is always a very small arc in terms of the radius,
usually from 20 to 60 minutes, the series will be rapidly converging
By means of Taylor's Formula, we may at once write
* ds ^ 2 eU* * 6 ds* '
(9*
We will use but the first three terms of this development, the fourth
term being used only in the largest primary triangles.
The derivation of the successive differential coefficients of / with
respect to s is the most difficult portion of this general development. U
s be supposed to vary, then / and s both must vary, and they are all im-
plicit functions of each other. These coefficients are therefore best
found geometrically, as follows : in Fig. 155,
■ /
Pic X55.
Let MB = BC — ds — differential portions of the line AB —sxn Fig. 154 ;
AD — — dl ^ change in AP {= /*) due to the change '\- ds'va s.
Let the anj^le PAB = z' and PBC = z'\ s" being greater than jbf' by
the convergence of the meridians shown by the angle AP'B.
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APPENDIX D, 697
The lines BD and CE are parallels of latitude through the points B and
C They cut all meridians at rijjht angles.
Since the triangle ABD is a differential one on the surface of the sphere,
it may be treated as a plane triangle, and we may at once write
df AD , , ,
the minus sign indicating that / and s are inverse functions of each other.
Differentiating this equation and dividing both sides by ds we obtain
d^l' , .dz' , .
Now the angle ds' is the angle AP' B, subtended by the arc BD with
radius BP\ But this arc, with radius ^iV gives the angle ds sin 2',
Therefore
BN
d*' = sin sds -^-^ = sin zds tan L' = sin z cot tds.
dz . , ^
-;- = sin 2 cot r m
ds
Substituting this in (11) we obtain
—^ = sm'« cot /^ • . (12)
Substituting these values in (9). we have
/ - / = — J cos / -♦- ij* sin' «' cot / 4- etc.
Now, replacing /, /", and z\ by L, L\ and Z\ we have
// — / = X cos Z' + W sin' Z' tan Z'. • , , (13)
Here s is expressed in arc to a radius of unity.
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698 SURVEYING,
Referring it now to the radius N, we have s ~ , where A' is the length
of the arc s in any unit, A^ being the length of the normal nm in Fig. 153,
given in the same unit.
Substituting these in (13), we have
,, , JCcofiZ* I ^ sin' Z' tan Z' , ,
^-^^—N^^'2 W* ' • • • • <'^J
This eives the difference of latitude in units of arc in terms of radius N.
But differences of latitude are measured on a sphere whose radius is
the radius of curvature of the meridian at the middle latitude. Since we
do not yet know the middle latitude, we can use the known latitude Z
and afterwards correct to -.
2
Changing to a sphere, whose radius is R\ and dividing by the arc of
I" in order to get the result in seconds, we have
Z' - Z = - 5Z = -ET^—iT cos if -h - -BT^^ — T, sin'Z' tan Z'. (15)
R arc I 2 R N arc i ^ ^'
If we let B = -iy-^—,T, and C = ^^"^ ^
iV arc i" 2ie'iV' arci"
we may write — 5Z = A'cos Z* '-5 + ^ sin» Z'C, ..... (16)
To reduce this to what it would be if the mean latitude had been used
we have to correct it for the ilifference in the radii of curvature, /?/,' and
R^, at the latitude L and the middle latitude respectively. If JZ be the
true difference ot latitude when R^ is used, and 6Z be the difference
when Rl is used, we would have
/IL:8Li:Rl iRmy
To reduce 5Z to JZ, therefore, we must add the quantity 5Z • J^
XI JP' ^^'-^
(l — ^^sin'Z)^'
whence ^ ^ tiU=/l^ll^"^^>ai.
(I — <H Sin'' L )i
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APPENDIX D. 699
Here dL is the diflference in latitude between one extremity of the
line s and its middle point, or dL = ^dL, as given in eq. (16), hence
\R^ / I - €^ Sin* L
-. _ le^ sin L cos L sin l"
Ti we put D = ' , —f-f, 1
*^ I — ^^ sm' L
the corrective term becomes
whence we finally obtain
^ JL = AT cos Z'-B'hJir sin* Z'-C-^(SLy.3. , . , , (D)
where dL is given by (16), or it is the value of the first two tcrrns in tlie
right member of this final equation. For distances less ilian 1? miles
the first term only may be used as giving the value of 6L.
The values of the constants B, C and Z> are given for every minute
of latitude from 23° to 65° in Appendix No. 7 of the U. S. Coast and
Geodetic Report for 1884. This Appendix can be obtained by applying
to the Superintendent.
For distances of 12 miles or less, using the first term only for 8L,
equation (18) becomes
/iL^ATcosZ' {B + AT cos rj?) + JC sin« ZC. . . . (E'l
THE DIFFERENCE OF LONGITUDE.
In the triangle APB, Fig. 154, the three sides and the angle at the
known station B are known. To find ^M = angle APB, we have,
therefore,
sin PA \s\n AB :: sin PBA : sin APB,
ox sin / : sin J : : sin a : sin ^M,
* In the U. S. Coast and Geodetic Survey Report for i8?4. Appendix 7, p.
326. this term is given with its denominator raised to the ^ power, and the
caDuiar values of D are computed accordinjfly. The develc-pnrient there given
is laborious and approximate, but the error is not more than o.ooi of the valu9
of this term, which is itself very small.
This was corrected in a new issue of this development by that department in
the Report of 1894, App. 9. , ^^^1^
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700 SURVEYING.
But - = tt; where N' is the normal Bn\ Fig. 154 ; and if we assume
that the arc s is proportional to its sine, we have
^M = -J^^^^-Pr, Cl»
iV^cosZ aicl
where AM is expressed in seconds of arc
If we put A= ^. l^ ^.u
Uus equation becoHies ^Af = -7; — • ••••••••• (Fj
^ cos L
In order to correct for the assumption that the arc is proportional to
its sine, a table of the differences of the logarithms of arcs and sines is
given in the U. S. C. and G. Report for 1884, p. 373, with instructions
Tor its use on p. 327.
THE DIFFERENCE OF AZIMUTH.
In the spherical triangle APB, Fig. 154, we have, from spherical
trigonometry,
cot «.- + .) = tan l(-^.»/t)^^f±^
COS ^L' — L)
But « = 180' — Z.
therefore cot Ki8o* - Z + z') = tan \{Z - O = tan i{JZ\
whence - Un \AZ = tan \AM TTrT^^x (^^51
cos \\L. — L,)
*Cbauvenet*s Spherical Trigonomelry, cq. (127).
t Increments of M are measured positively towards the west.
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APPENDIX D. 701
It will be seen that since the azimuth Z o( a, line is measured from the
south point in the direction S.W.N.E., the azimuth of the line BA
from B towards A (forward azimuth) is the angle PBA + 180** = Z',
while the azimuth of the same line from A is 180** — PAB = Z. Also,
that JZ = Z+ 180° + Z'.
Assuming that the tangents i^^Z and ^ J J/ are proportional to their
arcs, and putting Lm for the middle latitude, we have
-^Z= Jm-^^^ (Q
cos i/3L '
The U. S. Coast Survey Tables are based on the following semi
diameters:
fl = 6 378 206 metres,
^ = 6 356 584 •*
i*r a lb :i 294.98 : 293.98.
See Appendix No. 9, U. S. Coast and Geodetic Survey, Report for 1894,
for tabular values of constants and forms for reduction.
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APPENDIX E.
GEOGRAPHICAL POSITIONS OF BASE-LINES AND PRINCIPAL
MERIDIANS GOVERNING THE PUBLIC SURVEYS.
Since the adoption of the rectcingular system of public surveys. May 20,
1785, twenty-four initial points, or the intersection of the principal bases with
surveyin<^ meridians, have been brought into requisition to secure the cer-
tainty and brevity of description in the transfer of public lands to individual
ownership. From the principal bases townships 01 six miles square are run
out and established, with regular series of numbers counting north and
south thereof, and from the surveying meridians a like series of ranges are
numbered both east and west of the principal meridians.
Durino^ the period of one hundred years since the organization of the sys-
tem the following numerical and independent principal meridians and bases
have been initiated, to wit :
The first principal meridian divides the States of Ohio and Indiana.
having for its base the Ohio River, the meridian being coincident with 84^
51 of longitude west from Greenwich. The meridian governs the surveys of
public lands in the State of Ohio.
The second principal meridian coincides with 86° 28' of longitude west
from Greenwich, starts from the confluence of the Little Blue River with
the Ohio, runs north to the northern boundary of Indiana, and governs the
surveys in Indiana and a portion of those in Illinois.
The third principal meridian starts from the mouth of the Ohio River
and extends to the northern boundary of the Slate of Illinois, and governs
the surveys in said State east of the meridian, with the exception of those
rejected from the second meridian, and the surveys on the west to the
inois River. This meridian coincides with 89° 10' 30" of longitude west
from Greenwich.
The fourth principal meridian begins in the middle of the channel of
the mouth of the Illinois River, in latitude 38° 58' 12" north and longitude
90^ 29' 56" west from Greenwich, and governs the surveys in Illinois west
of the Illinois River and west of the third principal meridian lying north of
the river. It also extends due north through Wisconsin and northeastern
Minnesota, governing all the surveys in the former and those in the latter
State lyin^ east of the Mississippi and the third guide meridian (west of the
fifth principal meridian) north of the river.
The fifth principal meridian starts fpom the mouth of the Arkansas
River, and, with a common base line running due west from the mouth ol
the Saint Francis River, in Arkansas, governs the surveys in Arkansas,
I
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APPENDIX E, 703
Missouri, Iowa, Minnesota west of the Mississippi, and the third guide
meridian north of the river, and in Dakota Territory east of the Missouri
River. This meridian is coincident with 90° 58' longitude west from
Greenwich.
The sixth principal meridian coincides with longitude 97° 22' west
from Greenwicli, and, with the principal base-line intersecting it on the 40th
degree of north latitude, extends north to the intersection of the Missouri
River and south to the 37th degree of north latitude, controlling the surveys
in Kansas, Nebraska, that part of Dakota lying south and west of the Mis-
souri River, Wyoming, and Colorado, excepting the valley of the Rio
Grande del Norte, in southwestern Colorado, where the surveys are pro-
jected from the New Mexico meridian.
In addition to the foregoing six principal meridians and bases governing
public surveys, there have been established the following meridians and
bases, viz. :
The Michigan meridian, in longitude 84° 19' 09" west from Greenwich,
with a base-line on a parallel seven miles north of Detroit, governing the
surveys in Michigan.
Tne Tallahassee meridian^ in longitude 84° 18' west from Greenwich,
runs due north and south from the point of intersection with the base-line
at Tallahassee, and governs the surveys in Florida.
The Saint Stephen* s meridian^ longitude 88° 02' west from Greenwich,
starts from Mobile, passes through Saint Stephen's, intersects the base-line
on the 31st degree of north latitude, and controls the surveys of the south-
ern district in Alabama and of the Pearl River district lying east of the
river and south of township 10 north in the State of Mississippi.
The Hunt sville meridian, longitude 86° 31' west from Greenwich, extends
from the northern boundary of Alabama as a base, passes through the town
of Huntsville, and governs the surveys of the northern district in Alabama.
The Choctaw meridian^ longitude 89° 10' 30" west from Greenwich,
passes two miles west of the town of Jackson, in the State of Mississippi,
starting from the base-line twenty-nine miles south of Jackson, and termi-
nating on the south boundary of the Chickasaw cession, controlling the
surveys east and west of the meridian and north of the base.
The Washington meridian, longitude 91° 05' west from Greenwich,
seven miles east of the town of Washington, in the State of Mississippi, with
the base-line corresponding with the 31st degree of north latitude, governs
the surveys in the southwestern angle of the State.
The Saint Helena meridian, 91° ii' longitude west from Greenwich,
extends from the 31st degree of north latitude, as a base, due south, and
passing one mile east "f Baton Rouge, controls the surve)«s in the Greens-
borough and the southeastern districts of Louisiana, both lying east of the
Mississippi.
The Louisiana meridian ^ longitude 92° 20' west from Greenwich, inter-
sects the 31st degree north latitude at a distance of forty-eight miles west
of the eastern bank of the Mississippi River, and, with the base-line co-
incident with the said parallel of north latitude, governs the surveys in
Louisiana west of the Mississippi.
The New Mexico meridian, longitude 106° 52' 09" west from Green*
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704 SURVEYING.
wich, intersects the principal base-line on the Rio Grande del Norte, about
ten miles below the mouth ot the Puerco River, on the parallel of 34° 19'
north latitude, and controls the surveys in New Mexico, and in the valley
of the Rio Grande del Norte, in Colorado.
The Great Salt Lake meridian, longitude 111° 53' 47" west from Green-
wich, intersects the base-line at the corner of Temple Block, in Salt Lake
City, Utah, on the parallel of 40° 46' 04' north latitude, and governs the sur-
veys in the Territory of Utah.
The Boise meridian^ Icwigitude 116° 20' west from Greenwich, intersects
the principal base between the Snake and Bois^ Rivers, in latitude 43° 26'
north. The initial monument, at the intersection of the base and meridian,
is nineteen miles distant from Bois6 City, on a course of south 29° 30' west.
This meridian governs the surveys in the Territory of Idaho.
The Mount Diablo meridian^ California, coincides with longitude 121*^
54' west from Greenwich, intersects the base-line on the summit of the
mountain from which it takes its name, in latitude 37° 53' north, and governs
the surveys of all central and northeastern California and the initire State of
Nevada.
The San Bernardino meridian, California, longitude 116° 56' west from
Greenwich, intersects the base-line at Mount San Bernardino, latitude 34°
06' north, and governs the surveys in southern California lying east of the
meridian, and that part of the surveys situated west of it which are south
of the eighth standard parallel south of the Mount Diablo base-line.
The Humboldt meridian, longitude 124° 11' west from Greenwich, inter-
sects the principal base-line on the summit of Mount Pierce, in latitude
40*^ 25' 30" north, and controls the surveys in the northwestern comer of
California lying west of the coast range of mountains and north of township
5 south of the Humboldt base.
The Willamette meridian is coincident with longitude 122° 44' west
from Greenwich, its intersection with the base-line is on the parallel of
45° 30' north latitude, and it controls the public surveys in Oregon and
Washington Territory.
The Montana meridian extends north and south from the initial monu-
ment established on the summit of a limestone hill, eight hundred feet high,
longitude iii° 40 54" west from Greenwich. The base-line runs east and
west from the monument on the parallel of 45^46' 27" north latitude. The
surveys for the eniire Territory of Montana are governed by this meridian.
The Gila and Salt River meridian intersects the base-lme on the south
side of the Gila River, opposite the mouth of Salt River, in longitude 112°
15' 46" west from Greenwich, and latitude 33° 22.57" north, and governs
the public surveys in the Territory of Arizona.
The Indian meridian intersects the base-line at Fort Arbuckle, Indian
Territory, in longitude 97° 15' 56" west from Greenwich, latitude 34^31'
north, and governs the surveys in that Territory.
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APPENDIX F.
Note. — The following instructions, issued by the Mississippi River Commission
in 1891. embody the continuous experience of some twenty years' work on the
United States surveys of the Great Lakes, and of twelve years* work on the
survey of the Mississippi River, and are believed to represent the best practice
in secondary triangulation, precise leveling, topographic and hydrographic
work. J. B. J.
May^ 1893.
INSTRUCTIONS FOR SECONDARY TRIANGULATION, PRECISE
LEVEL, AND TOPOGRAPHICAL AND HYDROGRAPH-
ICAL FIELD WORK UNDER THE MISSIS-
SIPPI RIVER COMMISSION, 1891.
INSTRUCTIONS FOR SECONDARY TRIANGULATION.
Locating stations. — In locating stations it is desirable to fix them at
such points as give good conditioned triangles. The smallest angle in any
triangle should never be less than 30 degrees, and but few of these should
be permitted to enter into the system. The triangles should lie in such a
way that pointings can be made from any station to the stations immedi-
ately above and below on the same side of the river. That is, blind lines
should always be avoided. Other things being equal, stations should be
set where they can be readily found and where they will not be disturbed.
Reading angles. — The angles will be read with T. & S. theodolites,
Nos. I ana2. The instruments will be mounted firmly and protected from
sun and wind when in use. The value of the angle will be determined by
eight combined results read as follows :
Fig. xjfi.
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The instrument being at A, carefully leveled and in good adjustment.
With the vertical circle to the right, or telescope direct, and lower motion
fixed, point successively to A i, 2, 3, and 4, recording the reading of both
micrometers for each pointing. This eives a positive result for each angle.
Then point to A 4, 3, 2, i, and record readings as before. This gives a
negative result for each angle. A mean of the two gives one combined
result. The readings in a positive and negative direction will eliminate
twist of station or instrument, provided the readings occupy but a short
period of time during^ which the twist, if any, is uniform.
For the next conibined result. The telescope will now be reversed, that
is, revolved throujjh, leaving the pivots in the same wyes, and the whole
will be revolved \%o degrees in azimuth. The vertical circle will then be
on the left ; the limb will be shifted 22i degrees, and the stations will be
read forward and back as before. The notes for this series will be headed
circle left. Reversing the telescope will eliminate errors of collimation,
small level errors, and ineauality of pivots. Shifting the limb so as to read
the angles at equal intervals around the circle will eliminate periodic errors
and errors of graduation.
The same prop^ramme is followed until all the results are obtained, the
limb bein^ shifted and the telescope reversed after each combined result.
The micrometers should be adjusted so the run will be nearly zero. This
should, however, be tested at the beginning of each day's work, and entered
in the note-book.
Closing triangles, — The error in closing a triangle should rarely reach
and never exceed 6 seconds, and the average closure should be mucn below
this. This will require great care in the centering of instruments and tar-
gets. A discrepancy of one-third of an inch will give an error of a second
in a distance of i mile. A transparent cloth, phaseless target will be used,
the size varying with the length of triangle sides.
Base lines, — Base lines will be measured at intervals of about 75 miles.
This will be done with the 3oo-foot steel tape. The line should be carefully
staked out, and its grade determined instrumentally. Supporting stakes
will be driven at intervals of 30 feet. The stakes marking the extremities
01 each tape will be firmly set and free from any disturbing influence due
to tension of tape or otherwise. On these tapes strips of zinc will be fas-
tened and remain until the whole measurement is completed. The tem-
perature of the tape will be determined by three thermometers placed near
' the ends and in the middle of the tape. They will be attached to suitable
supports and placed with their bulbs near the tape when measurements arc
bemg taken. Observers must be careful to keep sufficiently far away so as
not to affect the thermometers.
The tape will be suspended in hooks at intervals of 30 feet, and attached
in such a way that it may swing freely and eliminate friction as far as prac-
ticable. The tension of the tape will be kept uniform while measuring by
attaching a weight of 16 pounds. The extremity of each tape leneth will
be marked on the zinc strip with a fine line and suitably numbered. The
preservation of these strips furnishes a ready means of comparison of each
tape length at any fuiure time.
The line should be measured two or more times, with a discrepancy
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APPENDIX F.
707
when reduced of not more than one in 250,000. This can readily be done
if measurements are made on cloudy days or at night.
Observations for azimuth, — The azimuth of each base line will be de-
termined by observing;, with a triaiio^ulation instrument, two closely circum-
polar stars at elong^ation on two different nights.
The instrument and light should preferably be at the extremities of the
base or a triangle side. The following order of observing will be used : *
First.
Second.
Third.
Fourth.
Circle right.
Point to light.
Point to star and note
time.
Read level direct and
reverse.
Point to star and note
time.
Point to light.
Shift limb 45 degrees.
Circle left.
Point to light.
Point to star and note
time.
Read level direct and
reverse.
Point to star and note
time.
Point to light.
Shift limb 45 degrees.
Circle left.
Point to light.
Point to star and note
time.
Read level direct and
reverse.
Point to star and note
time.
Point to light.
Shift limb 45 degrees.
Circle right.
Point to light.
Point to star and note
time.
Read level direct and
reverse.
Point to star and note
time.
Point to light.
On the second night repeat this programme, starting with a reading of
limb 45 degrees greater than the last reading of previous evening.
It will probably be found most convenient in these observations to use
Polaris, 5 Ursae Minoris, A Ursae Minoris and 51 Cephei.
The time will be determined by observing the meridian passage of hi-h
and low stars.
Stone line bench-marks, — At intervals of about 3 miles along the river,
lines of pipe and tile marks will be set for future surveys.
These lines will be numbered and located about as shown on maps on
file in this office.
The marks nearest the river will be far enough back to be safe from
erosion for many years ; the others will be half a mile farther back.
In cases where the bluffs are near the river the rear marks may be
omitted. The marks will preferably be placed at property corners, along
public roads, or on property lines, in places where they can be readily
found, and where they will not be liable to disturbance.
It is desirable to determine the azimuth and distance between the suc-
cessive marks on the same line when practicable. The marks should also
be as nearly in a line as the conditions of location above named will admit.
The marks will be connected directly with the secondary triangulation,
where practicable, by 3 pointings from 2 or more secondary stations, and
an equal number from the point to be located to 2 stations that will give a
fairly good triangle.
Where the points cannot be located directly from the secondary work,
a tertiary system may be used, starting and closing on a secondary line.
In this work the angles may be read with a good lo-second transit, and the
* If a mercury surface be used and alternate readings be taker- on the star and
on the image, the bubble readings may be dispensed with, as all errors from this
source are eliminated« J* B. J.
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7^^8 SURVEYING.
triangles should close within 1 5 seconds. A steel tape or chain may also be
used, where desirable, in locating the point which is farthest from the river.
Cutting timber. — Cutting timber to clear the lines of sight or for material
with which to build stations should be avoided as far as practicable. Where
cutting is necessary, a strict account must be kept of the number of trees
cut, their size, and kind of timber.
Descriptions of stations. — A minute description of each station will be
made and entered in notebook kept for that purpose. This description will
be complete for each station, and will show what the Geodetic point is and
how marked. Its location willi reference to surrounding objects will be
shown by an accurate sketch giving azimuth and distance to bearing trees,
houses, or other prominent objects.
A similar record will also be kept of the stone line marks.
INSTRUCTIONS FOR PRECISE LEVELS.
I. Before commencing operations the constants of the instruments will
be determined. The most important of these is the value of one division
of the level tube. This can best be determined by means of a level trier.
It can also be determined in the field as follows :
Set up the instrument firmly, if possible mounting it on a wooden post,
or, better still, on a stone pier. Set up a rod in its tripod at such a dis-
tance that it can be distinctly read through the telescope. The distance
should be at least 50 metres, or if the air is very still too metres, and
should be carefully measured. Adjust the instrwnent carefully, taking
such length of bubble in the level tube that its ends will be about the
middle or tenth graduated line on each side. Direct the telescope to the
rod, and by means of the elevation screw cause the bubble to run to near one
end of the level. Carefully note the 'position of the three wires on the rod
and the reading of the level. Now, by means of the elevation screw cause
the bubble to run to near the other end of tbe tube, and note the reading
of the wire and bubble as before. One result for value of i division of level
can then be obtained. This operation should be repeated 10 times. The
elevation of the rod should be changed occasionally between sets, in order
to avoid estimating the same part of the same centimetre on the rod. It
will be sufficient to run the bubble 5 divisions each side of its central
position.
If
k = distance from instrument to rod»
d, d} = distance through which eye and object ends of bubble move when
run from near eye end to near object end,
= amount of displacement of bubble between 2 readings.
r, r' — corresponding means of 3 thread readings on rod, and
V = value of i division of level in seconds of arc
Then
_ 2(r'-r)
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APPENDIX F. 709
2. With the value of i division of the level, tables will be constructed
showing the correction to be applied to a rod- reading for an observed in-
clination of the level, and for a distance determined by interval between
extreme threads.
If the level-bubble is well ground, equal displacements of the bubble,
say of 2 divisions, will correspond to equal displacements on the rod.
3. Before using the level, or determining its value, the fastening of the
tube in its case should be examined. One end should be clamped down
just tight enough to prevent the tube from moving easily, but not tight
enough to strain the glass. The other end should be lightly clamped so
that the tube may be free to expand and contract with temperature changes.
The cotton packing at the ends should not exert a lateral strain on the
tube. All level tubes will be numbered and have their numbers marked
upon them.
4. In order to determine the inequality in the telescope rings, the instru-
ment should be mounted on a stone pier or other firm support and care-
fully leveled. The level should be carefully adjusted and the instrument
clamped to prevent its moving in azimuth. Now, with the eyepiece of the
telescope over the elevating screw, note the reading of the bubble when
level is set on telescope, both in direct and reversed position. Now reverse
the telescope in the wyes, and read the level as before. Several sets of
observations should be made.
Let b,b^ ^ inclination of telescope as denoted by means of level readings
with telescope direct and reversed, then the inequality of rings/ =
4
Sixteen determinations of the value of p of two instruments in use on
the lake survey gave probably errors of ±o".046 and ±0 .041.
The inequality may be expressed in seconds of arc if desired, but for pur-
poses of computation is best expressed in terms of level divisions, as it can
then be combined directly with the error of adjustment of level.
5. The centering of the object glass will be examined. This may be
done as follows :
Draw out the eyepiece until the threads are no longer visible. Direct
the telescope upon some well-defined object, and while looking at it rotate
the telescope in its wyes. If the object remains steady, the object glass is
sufficiently well centered. Should the object appear unsteady, the fault
can only be remedied by a maker. The objective should be firmly screwed
into the telescope.
6. The values of the wire intervals will be determined as follows : Set
up a rod at carefully measured distances of 10, 20, 30, to 100 metres from
the instrument. Read the rod ten times at each distance. The rod may
be altered in elevation, the level may be caused to change, and the tele-
scope may be rotated 180 degrees (inverted) in order to change the position
of tne threads on the rod.
Taking the mean of the ten observed differences of readings of the
extreme threads at each station occupied by the rod, a table will be con-
structed giving in metres the distance of the rod from the instrument for
any observed difference of reading between extreme wires.
7. Unless the rods used have been previously compared with some
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7^0 SURVEYING,
known standard, they will be compared with each other and their relative
lengths determined. This may be done by establishing two fixed points,
or two foot plates, at equal distances from the instrument and differing in
elevation about 2.7 metres. The distance should be about 10 metres. De-
termine the difference of elevation of the noints by reading each rod on
each point. A comparison of the resulting differences of eleva^tion will give
relative lengths of metres on rods. Ten measurements with each rod will
be determined. The elevation of the instrument will be slightly changed
between each set in order to eliminate errors in estimating the millimetres.
Each rod will be numbered and have its number marked on it. The rods
should also be kept dry and provided with canvas covers to protect them
while being carried to and from work.
The distance of the zero graduation above the steel spur on which the
rod stands will be well determmed. This may be done with a right angle
triangle and rule. It may also be determined by means of another levehng
rod, the graduations of which commence at the foot of the rod, by determin-
ing the height of the instrument above some fixed point and subtracting it
from the reading of the rod to be determined. The relative lengths of the
rods must be known.
Whenever a bench-mark is connected with in such a way that the rod is
not placed directly on the bench-mark, this quantity (a) enters into the com-
putation of difference of elevation.
8. Before commencing work at anytime all adjustments will be carefully
made.
{a) The telescope will be collimated by having a rod set up at a distance
of 50 metres and noting the position of the wires on the rod when the tele-
scope is normal and when inverted or rotated 180 degrees about its axis.
The collimation error of the mean of the horizontal thread must not exceed
1.25 millimetres at a distance of 50 metres.
{b) The horizontality of the horizontal wires will be examined by mov-
ing the telescope in azimuth so that the rod shall appear to move through
the field of the telescope. If the threads are horizontal the reading on the
rod will be the same, the position of the level, which should be closely
watched, remaining the same. If the threads are found to be not horizontal
they will be made so by turning the telescope a small amount in the wyes.
When the thread wires have once been made horizontal, small screvv-s
which abut against projection of wye above elevating screw should be so
adjusted that when they press against this projection the wires are horizon-
tal. If the vertical thread is then inclined, as shown by the plumb line
attached to the rod, it must remain so.
{c) To make the axis of the level parallel to the upper surface of the
rings, it is necessary to make the vertical planes passing through them
parallel (lateral adjustment), and to make them equally inclined to the hori-
zon (vertical adjustment).
To make the lateral adjustment, raise the clips fastening the level to the
telescope, and revolve the level about the telescope a short distance each
side of the venical. If the bubble runs in opposite directions when on
opposite sides of the vertical, the level is to be adjusted by means of the
opposing horizontal screws at one end of the level until such is not the case.
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APPENDIX F. 71 1
To make the vertical adjustment, raise one of the clips and read the
level in its direct position and also when it is reversed on the telescope.
The difference between the differences of end readings in each position is
four times the error of adjustment, and is to be corrected by the opposing
vertical screws at one end of the level case. The error of adjustment must
not be allowed to exceed two divisions of the level. Care must be taken
that the telescope rings are free from dust while adjusting the level. After
having made the vertical adjustment it will be necessary to examine the
lateral adjustment again, since making one of these adjustments affects the
other.
(^) To make the level and vertical axis of revolution perpendicular to*
each other, loosen the small clamp screw at one end of the horizontal bar
fastened to the vertical axis and by means of the elevating screw rais« or
lower that end of the upper horizontal bar until the telescope can be rotated
180 degrees from any position and have the level reading the same in both
positions.
{e) To adjust the level attached to the rod, set up the rod in its tripod
in such a position that when a plumb line is attached to the small hook
near the top of the rod, the point of the plumb bob shall coincide with the
point of a small cone attached to the rod near its foot. Now bring the level
bubble to the center by means of the leveling screws. In making this ad-
justment the rod should not be exposed to the wind, as the plumb line is
influenced thereby. This adjustment will be made at least once each day.
Each time that the instrument is placed on a station, its axis will first be
made vertical by means of the leveling screws in such manner that the tele-
scope may be turned around the horizon without the bubble of the level
running a great number of divisions. The telescope is finally made hori-
zontal by means of the elevating screw. The inclination at the moment of
observing must not ordinarily exceed three divisions of the level, and never
five divisions.
The instrument when in use ought always to be sheltered from the sun
and wind. It is carried from station to station without being dismounted,
but the level should be taken off and carried in the hand. The small clamp
screw at the end of horizontal bar, and the large screw which fastens the
instrument immovably to the tripod, should both be turned tight before
moving the instrument.
The rods must be placed on the plates which accompany them and held
in a vertical position as indicated by the spherical level attached. It is
advisable to always use the same rod with the same foot plate. In placing
the foot plates great care should be taken that they be horizontal, on firm
ground, and not liable to change. The surface of the ground, if not firm or
level, should be removed.
The errors of adjustment will be determined at beginning and end o*
each series of observations ; that is to say, after having mounted the instru-
ment and before dismounting it, and in all. cases at least once each day.
If the instrument has been deranged by a jar the corrections must be deter-
mined anew.
The error of collimation will be determined by two readino^s of the rod
at a distance of 50 metres when the telescope is in its normal position and
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two when it is rotated 1 80 degrees in the wyes. The difference between thd
means of the two readings, atter being corrected for the inclination of the
level, must not exceed 2.5 millimetres at that distance, and commonly
should not exceed i millimetre. The error of the adjustment or the level
(inclination) will be determined by reading the level four times when direct
and four times when reversed on the telescope, reversing it between each
reading.
The error of adjustment must not exceed two level divisions, and com-
monly should not exceed one. All the details of the determination of the
errors of adjustment must be entered in the note book in their proper place.
It is always advisable to have the errors of adjustment as small as possible,
and necessary that they be well determined. The time of making these
determinations will be recorded in the note book.
In all work along the main line of levels each observer will duplicate his
own work by running over the line in opposite directions, preferably under
similar conditions as to illuminations, etc.
While connecting two bench-marks the order of using the rods will be
as follows :
/^ ^^ \
AAA A
.B.M. I u V a' I* a» 1' -
In the above figure let //', /' etc., represent the successive stations
occupied by the instrument. B. M. a\ a* etc., the positions occupied by
Rod I, and a, «* etc., the positions occupied by Rod 2. The instrument
having been set up at /, Rod i is placed on B, M. and Rod 2 at a, mak-
ing the distance I^a equal to /— ^. M. Rod i is then read, and imme-
diately afterward Rod 2. The time elapsing between these readings com-
monly will not exceed one minute and should not exceed 5 minutes. The
instrument is then carried to /^ and Rod i to «', the distances a — i'. and
I' — «\ being equal. Rod 2 will then be read, and immediately afterward
Rod I.
The instrument will then be taken to P and the rods read in the order
I, 2. Work will be continued in this manner until the other bench-mark is
reached. Rod i must be placed upon this bench-mark, which will be the
regular order if there have been an even number of instrument stations.
If there have been an odd number of instrument stations, at the last station
use Rod I for both backsight and foresight. While leveling the rate ot
progress in favorable weather will be about one kilometre per hour.
After having properly leveled the instrument at any station and having
made the vertical thread coiacide with the center line of the rod. the obser-
vation will be made and recorded in the following order : * First the level
will be read, the tenths of the division being estimated ; then the position
* It is preferable to keep the bubble in the center while threads are being read.
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APPEXDIX F. 713
of the threads on the rod will be read, the millimetres being estimated;
and finally the level will be read again. The o!>server will then read the
rod a second time to make sure that no error has been made. The recorder
will then take the differences between the readings ot the middle and extreme
wires to guard against errors, and if these differences denote any error the
observations must be repeated. If an error exists it will be shown by too
great a difference between the differences. This is a most important check
<md must not be neglected. These differences will also serve as a check
upon the distances between the instrument and rods.
The recorder should also check the level readings to make sure thai
errors of whole divisions have not been made. This may be done by
summing up the readings and noticing the length of the bubble. In rend-
ing the level by means of the mirror care should be taken that the position
of the eye is such that there will be no parallax. Such positions can be
determined once for all when the mirror is at its greatest angle of elevation,
by a second person reading the level directly while the observer finds the
position from which the reading of the level in the mirror is the same. The
notes will be kept in the form given in ncte books. When once a number
has been written down it must not be erased or made illegible. If wrong
a line will be drawn through it and the correct number written
above.
The lengths of sights taken will depend upon the condition of the
atmosphere, but the rods should always be near enough to be seen dis-
tinctly. It will be seldom that lengths of sights greater than 150 metres
can be taken. The backsight and foresight corresponding to any instru-
ment station must not differ in length by more than ten metres, and the
sum of the lengths of the backsights and foresights between any two bench-
marks should be equal.
Whenever it is necessary that the line of levels should cross a river or
other wide obstruction, a narrow place should be chosen. Firm points
should be set upon the two banks ; levels in good adjustment are set upon
posts about 10 metres from each bench-mark, and both levels go through
the same operation.
The error of adjustment is first accurately determined. — Call one of
the levels A. A first reads on the bench-mark near it, once with the
telescope normal and once with the telescope inverted, and then on the
rod across the river five times with the telescope normal and five times with
the telescope inverted. The error of adjustment of the level is again
accurately determined. The rod across the river will nted an extra vane.
B performs the same operation simultaneously. A and B change places
and repeat the observation at these new stations. The simultaneous levels
eliminate refraction, the change of station eliminates cuivature and small
instrumental errors. Unless good results nre obtained the levels should be
repeated. If but one level can be used the operation will be performed in
the same order, but the time occupied in crossing must be as small as pos-
sible. With a single Kern level this process has given for a river 815 metres
wide five results, the mean of which has a probable error of ±o"°».5. (Ohio
River, Cairo, III.)
Permanent bench-marks will be established at intervals of 3 miles along
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714 SURVEYING,
ihe river and 5 miles on lines connecting the river line proper with the
oihf r levels or bench-marks.
These bench-marks will consist of a thoroughly verified tile 4 inches by
18 inches by 18 inches placed 3 feet below the surface of the ground and
surmounted by a 4-inch wrought-iron pipe as a surface mark. The tile
siiould have time to settle before leveling to it. Both tile and pipe will be
suitably marked to designate the character of the point. In the center ol
the upper surface of the tile a copper bolt will be leaded, the upper surface
of which will be the point of reference. These bench-marks will be placed
where they can be easily found and where they will not be disturbed.
Property corners should be utilized where practicable.
In addition to the above, benches should be established on permanent
brick or stone structures by leading into them a horizontal copper bolt, with
the letters U^ S. P. B, M., and the number of the bench-mark cut near it.
A small hole in the center of the bolt will be the point of reference.
In connecting with a bench-mark if the bolt is vertical the foot of the
rod is placed directly Ujion it. If the bolt is horizontal in the wall of a
building or other structure, it may be best connected in the following man-
ner : Set up the instrument in such a position and at such an elevation that
the small hole in the bolt may be bisected by the middle thread without
displacing the level by more than five divisions, using the elevating screw
for making this bisection. Since the instrument can be raised or Towered
about two centimetres by means of the leveling screws, the instrument can
be placed in such a position by two or three trials.
Now bisect the bench-mark with the telescope normal and also inverted,
noting the reading of the level. Read the rod on the plate with the tele-
scope in both positions. It is necessary to eliminate collimation by invert-
ing the telescope, since the collimation of the middle wire is not the same
as that of the three wires. The quantity A (distance of zero above foot
of rod) must be taken into account when a bench-mark is connected with in
this manner. The distance of bench-mark from instrument must be
determined and recorded.
Whenever work is stopped at least two temporary bench-marks should
be established. These will consist of large nails or spikes driven their
entire length vertically into the base of trees, or in the tops of sound stumps.
When not in the vicinity of trees or slumps, wooden posts may be firmly
set in the ground with their tops flush with the surface and nails driven into
them. When near the river, temporary bench-marks should be set every
two kilometres. Every bench-mark will be fully described in a note book
kept for that purpose. Its position with reference to the most prominent
objects near it should be given by distance and direction. Public build-
ings, such as depots, court-houses, churches, etc., are the best positions for
permanent bench-marks. In a village or town several permanent bench-
marks should be established to secure some one against loss.
If a railroad is crossed the elevation of the foot of the rail will be
determined, and if leveling along a railroad, the elevation of the foot of the
rail at depots will be determined.
The elevation of the zeros of all water gauges and also the gauge bench-
marks will be determined.
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APPENDIX F.
;i5
The datum planes of cities along the line of levels will be connected with
aiid their elevations deduced.
Frequent connections will also be made with the United States Engineer
bench-marks between St. Paul and Grafton.
In reducing the observations the nearest tenth of a millimetre will be
retained. The distance will be taken out from the table to the nearest
metre.
The limit of discrepancy in closing a polygon will be —
3*"°* ^Distance in kilometres.
The distance referred to is the entire length of the polygon from bench-
mark I to bench-mark 2 and back to bench-mark i, and the limit of dis-
crepancy refers to the polygons between successive bench-marks. If the
discrepancy exceeds the prescribed limit, then the entire polygon must be
re-run one or more times, or until the difference of the means of the direct
and reverse results is within the limit.
The notes will be kept in the following form :
[Left-hand page.]
Difference
of threads.
307
ao6
4»3
BACK-SIGHT.
Thread
readinfiTA.
Mean.
Lc
Eye.
peL
Object.
Rod.
Remarks.
7.95
10.03
13.08
1001.7
II. I
II. I
13
1
Diftcrence
of threads.
187
186
373
FORESIGHT.
Level.
Eye. ' Object.
Rod.
[Right hand
Remarks.
page.l
Thread
readings.
Mean.
18.69
90.56
33.43
2055.7
II. 4
11.4
10
INSTRUCTIONS FOR TOPOGRAPHICAL AND HYDROGRAPHICAL FIELD
WORK.
The objects of the survey of the Mississippi River are to obtain sufficient
data for an accurate topographical and hydrographical map which may be
used in studying the physical characteristics of the river, planning improve-
42
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7l6 SURVEYING.
ments, and also serve as a basis for future surveys, by means of which the
chnnges in bed and banks may be ascertained and their causes and effects
studied. The importance of having the woric accurately done and the in-
formation embodied therein rehable is therefore apparent.
The experience derived in the surveys from Cairo to Donaldsonville,
covering a period ot several years, suggests the following instructions re-
lating to the scope of the work and the methods to be employed. Other
points will suggest themselves as the work progresses and new difficulties
are met with.
General instructions, — A record will be kept showing the daily progress
of the party. It will contain at the beginning the organization of the party,
and the names and rates of pay of all persons connected With it. It will
also give a detailed account of all occurrences of any importance which
may in any way be of use in reducing the work or in settling accounts.
At the beginning of each day's work each note book in use will givft
locality of work, date, name of observer and recorder, number of insiru*
ment used, and corrections, if any, to readings of distance and aiimuth.
In recording notes hard pencils will be used, and when an entry has
once been made it should never be erased. Where an error has been made
the record will be corrected by drawing a line through the first value and
writing the new value above it. Corrections that are made alter the work
is done sliould be marked with the date of the change and the name of the
person making the change.
All notes should be so full and plain that they could be readily reduced
by one who has not seen the ground. This will require careful attenrjcn
to details which may seem of trifling importance in the field.
All available information concerning the river and its adjacent banks
which will aid in the proper representation of the characteristic feature-s on
the map or be valuable in the study of their changes will be fully noted.
Local names of bars,. bends, streams, or other features will be carefully
noted and the proper spelling of all names to appear on the map will be
ascertained.
Permanent marks, as reference points for future surveys, will b* estab-
lished at intervals of about 3 miles along the river. There will be two on
each side of the river nearly in a line, normal to the stream. The two
nearest the river will be placed where they will be safe from the erosion of
the banks for 20 years or more, and the others will be a half mile further
back. Where the bluffs are near the river the outer marks maybe omitted.
These marks should, when practicable, be placed near roads, property lines,
or other places where they can be easily found at any time. The marks
will consist of flat tiles bearing wrought iron pipes (see instructions for
secondary triangulation), the tops of which should project not less than a
foot above the ground.
Note books will be fully indexed at the end of each day's work. Elacli
note book will be marked on the outside with a title giving locality of work,
date, names of chief of party and observer. All note booKs will be entereil
on the office files and properly numbered as soon as parties return from the
field.
The chief of party being responsible for the accuracy of the work done.
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APPENDIX F, 7^7
should see that the work of each member of the party is properly checked
and fully covers the ground required.
Tertiary triangulation. — Where the secondary stations are more than
3 miles apart a tertiary system will be carried giving points on either bank
at intervals of a mile or less. This system will begin on a triangle side of
the secondary system or a carefully measured base, and all of the available
secondary stations will be used in the tertiary chain. The tertiary work
will also close on a line of known length as a check on its accuracy. The
discrepancy should not exceed i in 3,000. The system should be laid out and
the angles read in advance of the topographers, so that the azimuths and
lengths of sides can be used in checking stadia work.
The station point may be marked by a pole 2 inches in diameter stuck
into the ground, and bearing a red and white flag to distinguish it from the
ordinary soundinjj flags. A strip of white cloth wrapped near the bottom
of the pole will admit of the pointings being made so low down that errors
arising from disturbance of the pole by the wind will be inappreciable.
For observing, the instrument may be placed on an ordinary tripod cen-
tered over the hole after the pole has been removed.
The angles should be read with a lo-second instrument in good adjust-
ment, and should be repeated at least three times on different parts of the
• line to check errors of reading.
It is desirable to have the first series read on azimuth. Having pointed
to the first station, read to all of the others in succession. Pointings should
also be made to all of the sounding flags in the vicinity, as well as promi-
nent objects on land, such as chimneys, houses, etc., the location of which
will serve to check the topographical work.
For the .second series slip the lower limb 60 degrees and read to the
stations in the opposite order from the first series. Slip the limb the same
amount a^ain and read the third series.
The river ends of the stone lines will be made points in the tertiary
system, and whenever practicable the stones should also be located trigono-
metrically.
Tertiary points which are likely to remain undisturbed for some time
should be plainly marked with a strong stake 2 feet high, the number of the
point, the initials of the observer and date being marked on it with red
chalk.
Topography, — The detailed topography will cover a belt on each side
of the river, which, in wooded country or on the bluffs, will be about one-
half to three-fourths mile wide and in open country may reach about \\
miles. In this area there will be located, with transit and stadia, all points
needed to plat accurately the important features on a scale of i :io,ooo. In
all work the scale of the plat should be borne in mind, so that only such
points be instrumentally located as can be readily platted.
Beyond the above limits outline surveys will be made defining streams,
lakes, and the foot and main crests of bluffs with approximate elevations of
same within a limit of 10 miles of the river. This work will be run with
the transit or compass and stadia, and will frequently be connected v.ith
the detailed topography.
Within the limits of the detailed area there will be located the top and
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7'^ SURVEYING.
bottom of the river bank proper, the shore line of islands and bars, the
banks and water lines of all waterways and lakes, with elevations of their
water surfaces and depths, the points where the slope of the ground changes
either in direction or inclination, the limits of rock ledges, the approximate
limits and kinds of cultivation, and forests, roads, levees, fences, houses,
etc., and in fact everything that may be necessary to a truthful representa-
tion of the section surveyed.
A sufficient number of elevations will be determined on the bottom lands
to admit of putting in contours 5 feet apart. In a wooded area this will
require cross sections at intervals of 500 metres or less. These should pref-
erably be the continuation of lines sounded across the river. The space
between the lines should also be examined and if any important features
are found they should be located.
When the trees are too close together to admit of long sights it will l>e
more expeditious and sufficiently accurate to use the compass needle for
obtaining the direction, as it will then only be necessary to set up at alter-
nate stakes.
The bluffs within the detailed area will be shown by contour lines 20
feet apart, the bluff curves being all some multiple of 10. The bluffs in the
outline area may be shown by hachures.
Boundary lines, such as State, county, township, etc., coming within the
limits of the survey, will be carefully located.
Section or township comers, where they are well identified, will also be
connected with.
Great care must be taken in running out the stone lines. The azimuth
of the lines must be accurately determined, and all distances will be care-
fully read, both forward and back. Each stone will be occupied instru-
mentally, and when practicable, the azimuth from one stone to the next will
be read, and readings will be made to surrounding objects both as checks
on the located positions of the stones and to aid in finding them in future
time. A careful sketch and minute description will be given of each bench-
mark thus located.
All sounding flags, water gauges, and bench-marks will be located.
In running the main transit line along the shore sufficient check shots
will be made to known points on the opposite shore, to prove the accuracy
of the positions given for the transit stakes. Such check shots should in
fact be made use of in all parts of the work, so that errors of azimuth and
distance may be detected and located. They also furnish means of correct-
ing errors of position if any occur.
The error of level carried with the transit should never exceed one foot
for the lonj^est distances. In good work the discrepancies will rarely reach
0.5 feet. The work will be frequently checked by starting and closing on
points whose elevations are known.
Distances and vertical angles between transit stakes will be read from
each end of the line. On transit lines the distances between stakes, read
with stadia, should never exceed 500 metres at a single reading unless they
can be checked by intersections or other means. Single shots to distant
objects may be read as far as the figures of the rod are distinguishable.
The transit stakes of each observer will be numbered consecutively in
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APPENDIX F,
719
the same reach and each stake will be marked with its proper number and
the initial of the observer so that it can be readily identified when connected
with by others.
Careful sketches will be made in the field of the entire area surveyed
and the located points will be indicated on the sketches and numberea to
correspond with the pointings in the notes so there will be no difficulty in
connecting the points properly on the field plats. The character ol the
immediate river bank will be frequently notecl so as to show whether it is
rock in place, loose rock, sand or silt, steep or sloping, caving or stable.
Checks on azimuth and elevation should be frequent, and when obtained
should be marked in the notebook in such a way that the amount of error
will be plainly shown and where the correction should be applied. Notes
that are not full in this particular will always be open to suspicion which
will throw doubt on the observer's honesty and the reliability of his work.
Discrepancies in closing on triangulation points should never exceed
0° 05' in azimuth or i in 500 in distance. As a rule the discrepancies
should be far within these limits.
The notes will be kept in the following form on the left-hand page of
the notebook, the other page being reserved for reductions, sketcnes and
remarks.
[Left-hand page.]
[Vicinity of Phillips Landing, Nov. 3, 1883. Inst. Wurd, No. 154— Dis. Short i in 100, Az. Cor.
F. B. Maltby, observer. F. P. Gibbs, recorder.]
Objects.
No. •f
pointing.
Ver.A.
Ver. B.
Distance.
Vertical
Read.
Corrected.
angle.
L Everett
T
2
At A I.
95 »o
127 16
0 1
275 10
xoo
250
lOX
252
+0.10
Top of bank
-J-0.8
[Right-hand pa^e.]
Diff. of
elev.
+0.96
+ 1.9
Elev.
290.3
991.2
292.2
•^1.9 392.2
Ordinary Uvels. — There will be a line of levels run along each bank of
the river, the ordinary y level being used for that purpose.
All turning points will he numbered so that the topogra
ily identify the points connected with.
purpcjbc.
\ topographers can read-
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726
^VRVEYWd
The ordinary level lines will connect with all precise bench-marks in
their vicinity.
The errors of closure should not in any case exceed 0.2 feet for the long-
est intervals. The two lines will check on each other at intervals of not
more than 3 miles or in the vicinity of each stone line.
The elevations of stone-line bench-marks will be determined by dupli-
cate lines of levels, the discrepancies between which should not exceed
0.05 feet. The adopted elevation will be the mean of the two determina-
tions.
The elevations of all permanent stations near the river, except those on
the bluffs, will be determined. Bench-marks will also be established on
each bank at intervals of about a mile. These may be placed on buildings,
trees, or other permanent objects near the river. A careful description,
sketch of location, and corrected elevations of all bench-marks will be made
and entered in a book kept for that purpose. These notes should be so full
as to enable one not familiar with the ground to find the marks even after
the lapse of several years' lime.
All water gauges will be cojHiected with by duplicate lines of levels
from the nearest bench-marks and the elevations of their zero points entered
in the gauge book. The elevation of tlie zero should be tested whenever
there is a probability that the gauge has been disturbed.
The elevations of the water surface will be determined at the extremi-
ties of sounding lines at intervals of not more than 400 metres, and the time
of the observation will also be entered in the notebook. This, when cor-
rected for change of stage, as shown by the local gauge readings, will give
the slope, and also serve to check large errors in leveling.
Elevations of transit stakes, high-water marks, and surface of ground at
sounding flags will be determined whenever it is practicable.
Level notes will be kept in the following form on the left-hand page of
the notebook, the other page being reserved for sketches and remarks.
[R. B. near Grand Tower, December la, 1884. Inst. B. & B. 140 M. Greenwood, observer.]
F.S.
Stations.
T P. 113
O 51
T. P.
"4-
B. S.
4.206
Ht. Inst.
1.400
3.159
a.380
Elevation.
336 321
339- «7
337.368
3390W
River crossings for connecting the two lines of levels will be made by the
two observers taking ten simultaneous readings across the river in opposite
directions. Then the observers and instruments should change places and
repeat the observations.
The instruments should be in good adjustment, and when once focussed
for the long distance should not be changed until the observations are com-
pleted. The mean of the values thus determined will be taken as the true
value.
Hydrography* — A continuous record of the stage of the river will be
* For smaller rivers the entire survey, includino: location of soundings, may well
be done by the stadia. See paper by J. L. Van Ornum, C. E., in Journal of tJU
Association of Engineering Societies^ vol. xiv., p. 219.
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APPENDIX F, 721
tierived from a suitable gauge read three times a day, its zero being referred
to a known bench-mark as soon after it is set as practicable.
Sounding lines will be run normal to the ^ream at intervals of 250
metres, and the soundings on these lines will be as close together as prac-
ticable. These lines will be numbered consecutively.
A continuous longitudinal line passing through the deepest water on
each section will be sounded.
On all crossings there will be sufficient soundings to determine the least
channel depth between the pools.
As many soundings as practicable will be located by means of angles
read simultaneously between located points on shore, wiih two sextants in
the soundinj^ boat. Intermediate soundings can be interpolated by taking
them at equal intervals of time.
The character of the bottom will be frequently determined by means of
a tallowed lead.
In water less than 10 feet deep it is convenient to use a pole divided to
feet and tenths. A lo-pound lead is suitable in water from 10 to 40 feet
deep. In greater depths it is desirable to use a -lead weighing from 15 to
20 pounds.
A firmly twisted or braided Ijemp lead line three-eighths of an inch in
diameter snould be used. It should be marked with leather or cloth togs
at intervals of one foot, the lo-foot marks being made conspicuous. The
length of the lead line from the end of the lead to each lo-foot mark must
be tested at the bej^inning and end of each day's work and the result entered
in the notebook. The lead line should be accurately marked, so as to avoid
corrections as far as practicable.
The beginning of each line sounded will be headed in the notebook with
the number of the section. A description of the character of the banks
at intervals of about half a mile will also be entered in the notelK)ok.
The elevation of the water surface at the time of sounding may be
determined for each line by means of the levels as already described
under the head of ordinary levels.
The notes will be kept in the following forms :
Sounding in the vicinity of St. Louis, November /j, i88g,
[E. L. Harmann, G. W. Wisner, observers ; D. E. Perkins, recorder ; J. Stott, leadsman.]
• [Left-hand page.]
[Right-hand page.]
and stage. I **^'*''
Remarks.
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722 SURVEYING,
The angles read to locate soundings will be numbered consecutively for
each clay's work, and the soundings located will be marked with corre-
sponding numbers.
Computing and platting. — The coordinates of all tertiary stations and
stone-line bench-marks will be computed, and, together with the secondary
stations, will form the basis for platting the topographical detail. The re-
sults of these computations will be kept in suitable form and preserved for
future reference.
The work will be platted on a scale of i : 10,000. The field plats will be
26 by 30 inches in size, on which, near the center, will be printed a 12-inch
circle divided to 15-minute spaces to facilitate the platting of polar coordi-
nates.
Parallels and meridians, i minute apart, will be projected on the field
plats and shown by fine red lines properly numbered. From these the a
stations will be platted. As this is the ground work for subsequent detail
it should be carefully done and checked over to insure its accuracy. All a
stations, stadia stakes, and sounding flags and their elevations should be
marked on the plats in red ink before the detailed work is put in.
All of the detail must be carefully platted and positions verified by check
shots when such are available.
The contour lines and other outlines should preferably be put in by the
observer who located them in the field.
Field plats will be laid out in such a way as to show both banks of the
river with the adjacent topography on the same sheet whenever it is prac-
ticable. If the sheet is not large enough, plat the remaining work on a
new sheet rather than enlarge by pasting pieces to the first sheet.
Banks that are too steep to admit of drawing in the contours, and
abrupt banks of less than 5 feet in height, will be shown by hachures.
The elevations of water surfaces for each day's work will be plainly
written on the plats.
All field plats must be completed in the field at least far enough to de-
tect any instrumental errors in the field work before the platted area is out
of reach. Hard pencils will be used in platting.
Each field plat will bear a legend giving locality and date, the names of
the chief of the party, the observers and draftsman, the numbers and pages
of the notebooks from which the notes were derived, and any other infor-
mation that may be useful in the final reduction of the work. This data
should be noted as the work is platted.
Care must be taken at the edges of the sheets to have the detail on suc-
cessive plats join properly, and make sure that the ground is fully cov«red
by the survey.
Nomenclature, — The word lake will be confined to the larger bodies of
water, which are seldom, if ever, dry. They usually have a local name
which should always be noted. Smaller and temporary bodies of water
having no local names will be called ponds.
The word swarap will be applied only to ground which is covered with
a growth of grass, cypress, elbow brush, willows, or such other vegetation
as indicates that the area is generally wet, soft, or spongy.
The ternjg bayou, or creek, will be applied only to main watercourses
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APPENDIX F. 723
which connect lakes and swamps or other drainage areas with the river
and carry water to or from the latter, as the stage varies.
Minor swampy conduits will be called sloughs. This applies only to
such as are not designated by local names.
The character of the material composing the bars and banks of the river
will be frequently noted and carefully described.
The names of property owners or residents, of landings, wood-yards,
fields, patches of timber, islands, chutes, bends, bars, points, and other
local names necessary to a full description of the section surveyed will be
fully noted and entered on the field plats. The following signs and abbre-
viations will be used : secondary stations, © ; tertiary stations, A ; transit
stakes, [3 ; sounding flags, 0. Turning points in leveling notes will be
written T. P. ; bench-marks, B. M. ; temporary bench-marks, T. B. M. ;
and precise bench-marks, P. B. M.
On the field plats the precise bench- marks, with their numbers and ele-
vations, will be written thus: P. B. M. 27 O 218'. 032 ; stone-line bench
marks thus : B. M. ^S- 0219.23. in which the numerator is the number of
the stone-line, and the denominator the number of the stone on the line
reckoned from the outer stoue on the left bank.
The stone lines will be numbered consecutively up stream, beginning
with number i, near Cairo, III.
All elevations in the topographical work will be referred to the Mem-
phis datum plane.
To reduce elevations from the Cairo datum to the Memphis datum sub-
tract 13.13 feet from Cairo datum elevations.
The approximate mean Gulf level is 8.13 feet above the Memphis datum
plant.
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APPENDIX G.
THE OWNERSHIP OF SURVEYS. AND WHAT CONSTITUTES
A SURVEY AND MAP.*
A GREAT difference of opinion seems to exist among surveyors as to how
much of the information obtained and work done in mating a survey
should be furnished to the individual for whom the survey is made. It is
believed that some surveyors have mistaken notions as to what constitutes
a survey and map and as to the ownership of the same. Many surveyors
keep what are called private notes. All men doing business as surveyors
must keep notes of all surveys in a convenient form for ready reference.
The extent to which these notes are •* private," however, seems to the
writer not to have been fully comprehended by all surveyors, and hence has
arisen the difference of opinion mentioned.
The present article is an attempt to present a side of this question that
has not, so far as the observation of the writer has extended, been hereto-
fore fully considered. An endeavor has also been made to point out to the
young surveyor a line of action which it is believed he will find to his ad-
vantage to follow, as well as to that of the community in which he works.
In this discussion, the question of what constitutes a survey arises at
once, and the answer obviously depends on the object of the survey. The
discussion will be confined to land surveys; that is, surveys made for the
purpose of subdividing a large tract of land into smaller parcels to be
sold, or surveys made for the purpose of determining the boundary of a
tract the description of which is known, or surveys made to determine
the description when the boundaries are known.
The principle to be enunciated applies to any other survey as well,
be it railroad, canal, bridge, or topograpliical survey. Indeed, it is well
understood in all such surveys, but seems to be ignored by many engineers
having to do with land surveys.
A survey is "the operation of finding the contour, dimensions, position,
or other particulars of any pait of the earth's surface, . . . tract of
land, etc., and representing the same on paper.**
In making a survey it is necessary to set certain points, called monu-
ments or corners, and to determine a description of these points. These
items therefore become a part of the survey.
Then a map must be drawn. This map, to be a faithful representation
* By William G. Raymond, C.E., Professor of Cieodcsy, Road Engineering,
and Topographical Drawing, Rensselaer Polytechnic Institute. The original paper,
of which this is a slight modification, appeared in J he Polytechnic, the student
journal of the Rensselaer Polytechnic Institute, January, 1894.
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APPENDIX G. .7-5
of the ground and the work done, should, together with the notes, show
all of the items mentioned.
The object of establishing monuments or corners and describing them
is a double one, viz.: tlie marking on the ground of the boundaries of
the tract, and the securing of definite information as to the location of the
tract with reference to other points or tracts, so that from this information
the land may at a future lime be found.
The survey is not complete, therefore, till the corners are fixed, infor-
mation that will preserve their location obtained, and the same delineated
on a map and its accompanying notes.
The doing of all this, then, constitutes the survey. To whom belongs
the survey ?
It would appear to be evident that it belongs to the individual who
pays to have it made. It is not readily seen in what way the survey, or
any part of it, becomes the sole property of the surveyor.
The surveyor may keep a copy of his notes to facilitate his future work,
but he has not the shadow of a claim to a single note the time for taking
which has been paid for by his employer.
If his charge for his work is on a time basis, there can be no question
as to the correctness of the above proposition. If he takes the work for a
definite sum for the entire job, he may take as much lime as he likes and
as many private notes, but he is bound in honor to return to his employer
the complete survey, and if he does so, it is not obvious that the private
notes would thereafter be of great assistance to him in securing further
work, particularly when it is remembered that professional men of repute
do not bid against each other for professional work.
His reputation for accuracy and honesty will be afar more potent factor
in securing employment than any set of private notes fairly obtained.
It is true that a great many surveyors hold a different opinion, and pur-
posely return their maps ancl notes in such condition that, while they may
answer the purpose for which they are primarily made, they do not tell
the whole story, nor enough to make it easily possible for another surveyor
to re-locate the tract surveyed. When this is done, the person ordering
the survey does not receive what he pays for. Something is withheld. It
seems to need no argument to show tnat this is radically wrong.
But there is another reason for condemning this practice.
The correct and permanent location of all public land lines, as streets,
alleys, etc., as well as the permanent location of party lines between
private owners, is a matter of the gravest importance, and no information
that will at all serve to definit^y fix such lines in their correct positions for
all lime should be withheld from the owner who pays for the survey, be it
private citizen, municipality, county, or State.
The records of monuments and street lines made by a city engineer are
no more his private property than are the records in the offices of the
clerk, auditor, or treasurer the property of the individuals who hold office
at the time the records are made.
The correctness of the position assumed has been indicated by court
decisions. A great deal of laxity is shown in the conduct of offices of city
engineers and county surveyors.
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726
SURVEYING.
I
The methods of regulating the pay of these officers has doubtless had
much to do with this. It is not uncommonly the case that the surveyor
receives no salary, but is allowed to collect certain specified fees for work
performed, and this gives color to his claim that his work is private work
and belongs to him.
That this is not true concerning the public work he does, is believed to
be evident from what has preceded. That the records of work done for
rivate citizens are not the property of the public, needs no demonstration,
ut it is true that such work belongs to those citizens for whom it was
done.
The writer believes that a different policy should be pursued with
regard to these offices.
He believes that in every case such office should be a salaried one.
with such salaried assistants as may be necessary, and that certain lees
should be prescribed for performing the various kinds of work thnt the
surveyor may be called upon to do within the limits of the territory of the
political division whose servant he is. These fees should cover all work
connected with public construction or public or private land lines, and
should be returned to the public treasury.
Their amount may be regulated from time to time so that they shall
aggregate a sum sufficient to pay the expenses of the office. They should,
of course, not cover work of a private character, not having to do with land
lines. But the entire public is interested in the permanency of land lines,
and all records concerning them made by a public official should become
public property. The writer has had in the past some experience in this
class of work, and never declined to furnish a competitor with any infor-
mation in his possession that would help the competitor to arrive at the
truth in surveys he might have under way.
The writer believes that the permanency of land lines is too important
a matter to be subject to avaricious and jealous rivalry, and he believes
that all the surveyors in a given district should cobperate to preserve in
their correct places all lines within the district.
To this end the returns of every surveyor made to the owner should be
thoroughly complete. Maps made for filing as public records should be
so finished as to enable any surveyor to re-locate the land without the least
uncertainty as to the correctness of his work. That this is done in very
few instances is well known to every surveyor who has had occasion to
examine public records for data for surveys he has been called upon to
make.
Because of the fact that in most cases neither owners nor attorneys have
been fully posted, nor could they be expected to be, as to what constitutes
a complete description, sufficient for re-location, and because surveyors
have been willing to let matters stand as they were, great carelessness has
arisen in the practice of making and filing maps for record.
♦ While in some States good laws exist prescribing what shall appear
on a map before it will be received as a public record, in more States
• What follows IS a modification of some notes on this subject prepared by the
writer for the Technical Society of the Pacific Coast.
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APPENDIX G. • 7-7
there is nothing whatever to guide either owner, surveyor, attorney, or
recorder in the matter. In the county records in such States anvthing that
is made up of lines and figures, either drawn by hand, photo-iitnographed,
or simply printed with "rule" and type and labelled "This is a map," is
considered a sufficient basis for the correct description and location of the
property it purports to represent. The records are full of auctioneers'
circulars, manufa.. lured in a printing-office from information, coming from
nobody knows where, filed at the request of the auctioneer's clerk,
with no name of owner or other interested party attached, except as the
name of the auctioneer appears in the accompanying advertisement.
Further than this, these maps are frequently purposely distorted to create
a favorable impression of the property to be sold. Wide streets are shown
where only narrow ones exist, streets opened for the full width where
they have been opened for but half their width, rectangular subdivisions
that really may not be even parallelograms, etc., etc. Such maps as these
frequently form the only basis for the description and location of the
property they are supposed to represent. This circular business is bad,
very bad for those who buy ; but is the information given by these circu-
lars much worse than that furnished by many of the maps made by sur-
veyors and filed at the request of the owners ?
On these plats, if of "additions," we find lines indicating the boundaries
of blocks and lots, all of which blocks and lots. are numbered ; the names
of streets appear in neat letters ; a few dimensions, possibly all linear
dimensions, will be given ; the streets or blocks may be tinted with soft
and delicate tints, and the whole set off with an elegant border and title.
As an exhibition of the draughtsman's skill these maps are perhaps
valuable. As a source of information as to the location of the lines they
purport to show, they are worth about as much as the auctioneer's circular.
Perhaps they have a few more figures, and the presumption may be a
jittle stronger that the figures are correct.
Examine one of these maps closely. There will be found no evidence
that a monument has been set in the field ; not an angle recorded, though
the lines may cross at all sorts of angles ; and dimensions given that do
not agree among themselves, so that the angles cnnnot be calculated.
There will be found no name signed, except, possibly, that of the sur-
veyor, who thus, advertises either his stupidity or something worse. Let
us be kindly, and call it stupidity.
Frequently no monuments are set except small stakes 9*. the corners of
the blocks ; but the fact that even such stakes have been se^ is not recorded
on the plot.
One who is acquainted with the practice of surveyors in a eiven district
knows at what points to look for such stakes, and if they hove oeen set and
not pulled out to make room for a fence post or building, he may succeed
in finding them. Some surveyors have a practice of setting stakes a certain
distance away from the point the stake is supposed to mark, but no mention
of this fact appears on the map. In fact the map is so drawn that no one
but a surveyor who made it can write a description of any one of the par-
cels of land shown, nor correctly locate it on the grouncl. Furthermore,
the surveyor himself finds it impossible, after the lapse of a few years and
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SURVEYING,
the destruction of his •• private marks," to re-run any one of the lines exactly
as originally laid out.
It is easy to see to what this leads : impossible descriptions of property
giving opportunity for differences in judgment as to interpretation of what
was intended ; disputes as to position of party lines ; costly litigation and
expensive movement of structures begun or completed, and the actual
shifting of lines back and forth by different surveyors, or even the same
surveyor, honestly trying to locate the lines properly.
The writer has seen enough of trouble of this sort to indicate to him
that a radical change is needed in the field work and mapping of cities,
towns, and additions, not to mention farms and other tracts of land that it
may be necessary to lay out and describe.
So long as fallible man is responsible for the accuracy of surveys, maps,
and descriptions of properties, so long will there be errors ; but that it is
possible to greatly reduce their number by proper regulation, the writer is
fully persuaded. What we have been describing are not maps at all, or
at most they are very imperfect maps, and " What constitutes a map ? "
thus seems to be a very pertinent question.
A map of a city, town, or addition, or other tract of land, serving as a
basis for the description of property, should furnish all the information
necessary to the proper description and location of the various parcels
shown, and also of the whole piece. It should further show the exact
location of the whole tract to the lands immediately adjoining ; particu-
larly should this be done when an offset or angle in a street line occurs.
To accomplish these things there should appear on the map the following
items :
1. The lengths of all lines shown.
2. The exact angle made by all intersecting lines.
3. Tire exact position and character of all monuments set, with notes of
reference points.
4. The number of each block and lot.
5. The names of all streets, streams or bodies of water, and recognized
landmarks.
6. The scale.
7. The direction of the meridian, and a note as to whether the true or
magnetic meridian is shown — it should be the true meridian.
8. The angles of intersection made by the lines of adjoining property
with the boundaries of the tract mapped.
9. The exact amount of offset in lines that may extend from the outside
through the tract mapped.
10. A simple, complete, and explicit title, including the date and the
name of the surveyor.
Thus much to make the map valuable for description and location of
the property it represents.
Of course monuments will not be shown if none have been set, and very
frequently none are set, either from carelessness on the part of the sur-
veyor, or an unwillingness on the pan of the owner to pay their cost.
Monuments of a permanent character should be set at each corner of a
tract surveyed, and at least two, visible the one from the other, on the line
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APPENDIX G. 729
of each street. If these monuments are not placed on the centre lines of
the streets, they should be placed at uniform distances from the centre or
Croperty lines. If placed with reference to the centre line, they should all
e placed on the same side of the centre. In streets extending east nnd
west the monuments should all be on the north of the centre, or tliey
should all be on the south, and at uniform distance. In streets extending
norih and south the monuments should all be on the east of the centre, or
all on the west.
Uniformity in such practice saves a vast amount of time.
Monuments may be set at uniform distances from the block lines, in the
sidewalk area, and this is an excellent practice.
The stakes or monuments set at the corners of the blocks in additions,
or town sites, should never be the only stakes or monuments set in the
tract.
That the map may be reliable there should appear on it the following:
1. The certificate of the surveyor that he has carefully surveyed the
land, that the map is a correct representation of the tract, and that he has
set monuments (to be described) at the points indicated on the map.
2. The acknowledged signature of all persons possessing title to any of
the land shown in the tract, and, if possible, those ot adjoining owners.
3. If of an addition, the acknowledged dedication to public use forever
of all areas shown as streets or roads.
4. If a street of full width, whose centre line is a boundary of the tract,
is shown, the acknowledged signature of the owner of the adjoining prop-
erty, unless his half of the street has been previously dedicated.
It has been already stated that, in some Slates, a map may be filed at
the request of any person, and without signature.
This practice frequently leads to trouole. The writer knows of cases
in which owners of large tracts of land have had those tracts subdivided
and have taken land of adjoining non-resident owners for street purposes
without the consent or knowledge of those owners. When at a later day
the owners of the land so taken have objected and attempted to close half
of the street, trouble of a serious character has arisen. The same trouble
has occurred where streets have been run through narrow gores of land
and have subsequently been completely closed, leaving houses built on the
mapped property without outlet. Time and again have cases of this sort
come to the knowledge of the writer.
Having pointed out certain evils, it remains to suggest a remedy.
It lies m the enactment of a law in each State governing these matters.
There should appear on the statutes of every State a law explicitly
defining what shall appear on every map filed for reference, and making it
a misdemeanor to file a map that cioes not strictly conform to the dehni-
Xion.
In the absence of such laws it is believed that the young surveyor can
assist greatly in a much-needed reform, by following the principles
suggested in this paper as the correct ones, and avoiding the errors here
indicated.
It is hoped that those graduates of our engineering schools who drift
into this class of work will be guided by a higher principle than that which
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73^ SURVEYING.
actuates the surveyor who covers up his tracks, at the expense of his
employer, in order to secure a monopoly of the business of his locality.
The young surveyor can spend his energies to greater advantage in
devising new and better methods of work than in inventing ways for hiding
information that it has been endeavored to show belongs to his employer.
Certainly a thorough education should so broaden the young man's
views as to make it impossible for him to be controlled by those meaner
instincts which, indulged, lead ever to narrow his vision and prevent him
from perceiving the greater problems that continually present themselve5%
for solution.
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APPENDIX H.
INSTRUCTIONS RELATIVE TO MAKING AND FILING OF
TOWN, CITY. AND VILLAGE PLATS IN THE STATE OF
MICHIGAN.
Auditor General's Office, )
Lansing, Mich., 189.. j
By Act No. 309, Laws of 1887, important amendments have again |^een
made to Sections i and 2 of Chapter 32 of Howell's Annotated Statutes in
regard to the recording of town plats and the vacating of the same. These
amendments took effect September 28, 1887, and all plats made on or after
that date must be in conformity therewith.
1. In making the SURVEY it is required that " permanent monuments
shall be located in the ground at all angles in the boundaries of the land
platted, and at all the intersections of streets or streets and alleys, as sliown
on the map or plat, and when there are permanent objects in the vicinity
of such monuments the bearings and distances of such objects shall be
noted. The character of the monuments and the bearings and distances
of such witness points or objects shall be distinctly given in the most con-
venient manner on the plat." The exact position of the monuments should
by indicated on the plat by a small circle or cross.
2. If the plat be of a town, city or village, the full name of such town,
city or village must appear as the title or name of the plat ; if the land
platted be an addition to or a subdivision of a town, city or villa^^e already
platted, then let the title of the plat include, with the name of sucn addition
or subdivision, the name of the town, city or village, as the case may be,
of which such platted land is a subdivision, or to which it is an addition.
3. The plat must be on a scale showing not more than two hundred
feet to an inch, and on good muslin-backed paper, 18x24 inches in size ;
all certificates must be written or printed on the paper on which the plat is
made, and on the same side of the sheet.
4. The sections and parts of sections platted must be designated by
lines with appropriate letters and figures. In case of a subdivision of lots
or blocks of a previous survey, the outlines of the original or previous lots
or blocks so subdivided must be designated by lines w-hich must be marked
with appropriate letters and figures. This must be done in such a manner
as to show, without reference to the written description, the starting point
and the course and length of each of the outlines. Where any of the out-
lines are identical and co-terminus with the lines of a previous survey or
plat, it will be sufficient to give the destination of such outlines as given in
such previous survey or plat.
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732 SURVEYING.
5. The land platted is to be fully described in writing or printing ^^^xi
the paper on which the plat is drawn. This description must be so com-
plete that from it, without reference to the plat, the starting point can be
determined and the outlines run. In connection with the description
should be a short and simple form of dedication, which must be signed by
the proprietors and their wives, whose signatures must be witnessed, and
whose execution of the dedication must be acknowledged as deeds convey-
ing land are required to be witnessed and acknowledged.
6. There must be drawn upon the plat a plain designation of the car-
dinal points and a correct scale.
7. Wnere all the lots in any block are of trie same dimensions, it shall
be sufficient 10 mark the precise length and width upon one tier thereof,
but all gores, triangles, or other lots, which are not either squares or par-
allelograms, shall have the length of their sides plainly defined by figures.
8. The streets must be named or numbered and their course and width
designated. All public grounds and alleys must be properly designated.
9. The surveyor must certify that the plat is a correct one and that the
monuments described in it have been planted as therein described.
10. Detached parcels cannot be included in one plat, nor can more than
one plat be made on one sheet. Contiguous parcels owned by different
parties may be embraced in one plat, all joining in the execution and
acknowledgment ; it is not necessary to specify the particular parcels
belonging to each.
11. Before a plat shall be recorded, and before any copies are made
therefrom, the "original" must be forwarded to the Auditor General for his
approval ; if found, in his opinion, to conform to law, it will be endorsed
as approved and returned to the person sending it.
12. For the purpose of recording, an exact copy is to be made from the
original after it has been approved by the Auditor General, which copy
must have copied upon it the Auditor General's endorsement of approval
and then be certified as follows :
State of Michigan, )
County of ) * We Register
of Deeds, and Surveyor, hereby certify that
we have each carefully compared this copy with the original plat of
and that it is an exact copy thereof and of
[title of plat.]
the whole of such original map or plat.
Register of Deeds.
Surveyor.
13. The ** copy " so certified is to be delivered to the Register of Deeds
and Hied as the record, and must have the proper endorsement of record
made upon it. No "copy" can be received for record unless the Auditor
General's certificate of approval is copied thereon. The transcript from
the record that is to he hied in tlie office of the Auditor General should
have copied upon it the above certificate of comparison of the "copy " with
the original, the approval of the Auditor General, and the certificate of the
Register of Deeds as to the time, volume and page of the record. The foi-
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appeVdix //. 733
lowing certificate of comparison with the record must then be made upon
the transcript, which should at once be filed with the Auditor General :
State of Michigan, ) ^^
County of J * We Register
of Deeds, and Surveyor, hereby certify that
we have each carefully compared the annexed copy with the plat of
now of record in the office of said Register
[title op plat.]
of Deeds, and that it is a true transcript therefrom and of the whole of such
record.
Register of Deeds.
Surveyor.
14. The Register of Deeds is entitled to a fee of $2.00 for each plat
recorded by him, and each plat filed with the Auditor General must be
accompanied by the legal fee of $1.00 for the benefit of the State.
15. The foregoing is not intended to be a perfect manual containing all
that is embodied in the law, but to call attention to points in which plats
are most likely to be defective.
16. Every plat sent to the Auditor General, either for approval or for
filing in his office, should be accompanied by the name and postoffice
address of the person sending it, to insure proper return.
17. The law of 1887 is appended for convenient reference ; it should be
carefully studied and all its requirements observed, (Omitted in this
Appendix.)
18. Observe the requirements of Section 135, General Tax Law of 1893,
which is hereto appended.
19. The following resolutions were adopted by the Michigan Engineer-
ing Society at their annual meeting held at Ann Arbor in January, 1886,
which are worthy of attention as expressing the views of an association
composed of the most intelligent and competent surveyors of the State:
Firsts That the written description of the land platted should be clear and
distinct, describing it in as brief a manner as is consistent with accuracy,
so that there shall be no misunderstanding as to what land the plat is
intended to cover ; that the outlines of the plat itself shall be marked with
appropriate letters and figures corresponding wiih the written description
indicating the courses and length of those lines ; that if any lots lying
within the outlines of the plat are not intended to be considered a part- of
the plat they should not be numbered or lettered, and the fact that they
are excepted be noted in the written description. The courses and length
of the lines of such lots should be marked on the plat by appropriate letters
and figures, and the lots themselves marked on the plat as excepted.
Second, Use a short and simple form of acknowledgment. . . . Third,
Use but one unit of measurement in the plat. Ma-ke a diagram on the plat
of the scale used, with appropriate letters to show what the scale is.
Fourth, The four cardinal points be indicated in a simple manner by an
arrow ox fleur de lis, with appropriate letters. This we understand to be
intended merely to indicate in a general way the points of compass on the
map. Whenever practicable, give the courses in the written description
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734 BUkVEflNG.
and on the plat from the true meridian, and also, when practicable, that
the angle of intersecting lines be given on the plat. Fifth, That we esteem
it of the first importance that permanent monuments be located in the
ground at all important points in the plats, and that the character and loca-
tion of such monuments, by their bearing trees or points, be distinctly
given on the plat.
FORMS OF DEDICATION. DESCRIPTION, AND SURVEYOR'S
CERTIFICATE.
Auditor General's Office, )
Lansing, Mich., 189. . f
In response to frequent requests for forms of Dedication, Description
and Surveyor's Certificate to be observed in makine plats, the following
have been prepared ; while ihey are short and simple, it is believed they
meet the requirements of law.
Stanley W. Turner, Auditor General.
|3r" (The Attorney General advises that if the proprietor is a widower or bachelor the fact
should be stated in tlie dedication to account for the absence of the signature of the wife.)
DEDICATION.
Know all men by these presents ^ That we as
proprietor, and his wife, have caused the land
embraced in the annexed plat to be surveyed, laid out and platted, to be
known as and that the streets and alleys as shown
(Insert title of Plat.)
on said plat are hereby dedicated to the use of the public.
Signed and Sealed in presence ^/ ) Pt <5 1
•■•'•-''••■'•■•■■''-'•-'^ [L.S.]
State of Michigan, )
County of \^^' On this day of 189. .
before me, a Notary Public in and for said county, personally came the
above named and
his wife, known to me to be the persons who executed the above dedica-
tion and acknowledged the same to be their free act and deed.
Notary Public Co., Mich.
description of land platted.
The land embraced in the annexed plat of.
is described as follows :
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APPENDIX H. 735
surveyor's certificate.
I hereby certify that the plat hereon delineated is a correct one and that
permanent monuments, consisting of
(Describe the monuments.)
have been planted at points marked thus as thereon
(Using some symbol such as a small circle (O)^ or a (X)> or letters, or numerals, to indicate the
exact location.)
shown at all angles in the boundaries of the land platted, and at all inter-
sections of streets and alleys.
Surveyor.
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APPENDIX I.
RESTORATION OF LOST OR OBLITERATED CORNERS
AND SUBDIVISIONS OF SECTIONS.
(Circular issued by the General Land Office of the Department of the Interior,
October i6, i8g6.)
The increasing number of letters from county and local surveyors re-
ceived at this ofhce making inquiry as to the proper method of restoring
to their original position lost or obliterated corners marking the survey
of the public lands of the United States, or such as have been wilfully
or accidentally moved from their original position, have rendered the
preparation of the following general rules necessary, particularly as in
a very large number of cases the immediate facts necessary to a thorough
and intelligent understanding are omitted. Moreover, surveys having
been made under the authority of dififerent acts of Congress, different re-
sults have been obtained, and no special law has been enacted by that au-
thority covering and regulating the subject of the above-named in-
quiries. Hence, the general rule here given must be considered merely
as an expression of the opinion of this office on the subject, based, how-
ever, upon the spirit of the several acts of Congress authorizing the sur-
veys, as construed by this office, and by United States court decisions.
When cases arise which are not covered by these rules, and the advice of
this office is desired, the letter of inquiry should always contain a descrip-
tion of the particular corner, with reference to the township, range, and
section of the public surveys, to enable this office to consult the record.
An obliterated corner is one where no visible evidence remains of the
work of the original surveyor in establishing it. Its location may, how-
ever, have been preserved beyond all question by acts of landowners, and
by the memory of those who knew and recollect the true situs of the
original monument. In such cases it is not a lost corner.
A lost corner is one whose position cannot be determined, beyond
reasonable doubt, either from original marks or reliable external evi-
dence.
Surveyors sometimes err in their decision whether a comer is to be
treated as lost or only obliterated.
Surveyors who have been United States deputies should bear in mind
that in their private capacity they must act under somewhat different
rules of law from those governing original surveys, and should carefully
distinguish between the provisions of the statute which g^uide a Govern-
ment deputy and those which apply to retracement of lines once sur-
veyed. The failure to observe this distinction has been prolific of
erroneous work and injustice to landowners.
736
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APPENDIX /. 737
To restore extinct boundaries of the public lands correctly, the sur-
veyor must have some knowledge of the manner in which townships were
subdivided by the several methods authorized by Congress. Without
this knowledge he m:y be greatly cmbarrc.ssed in the fie!d, and is liable
to make mistakes invalidating his work, and leading eventually to serious
litigation. It is believed that the following synopsis of the several acts
of Congress regulaling the surveys of the public lands will be of service
to county surveyors and others, and will help to explain many of the dif-
ficulties encountered by them in the settlement of such questions.
Compliance with the provisions of Congressional legislation at dif-
ferent periods has rcsvlted in two sets of corners' being estabHshed on
iozvnship lines at one t'me; at other times three sets of corners have been
established on range I'.ncs; whi e the system now in operation makes but
one set of corne.s en tozitiship boundaries, except on standard lines — i.e.,
base and correction lines, and in some exceptional cases.
The following brief explanation of the modes which have been prac-
tised will be of service to all who may be called upon to restore oblit-
erated boundaries of the public land surveys:
Where two sets of corners were established on township boundaries,
one set was planted at the time the exteriors were run, those on the
north boundary belonging to the sections and quarter sections north of
said line, and those on the west boundary belonging to the sections and
quarter sections west of that line. The other set of corners was estab-
lished when the township was subdivided. This method, as stated, re-
sulted in the establishment of two sets of corners on all four sides of the
townships.
Where three sets of corners were established on the range lines, the
subdivisional surveys were made in the above manner, except that the
east and west section lines, instead of being closed upon the corners pre-
viously established on the east boundary of the township, were run due
east from the last interior section corner, and new corners were erected
at the points of intersection with the range hne.
The method now in practice requires section lines to be initiated from
the corners on the south boundary of the township, and to close on ex-
isting corners on the east, north, and west boundaries of the township,
except when the north boundary is a base line or standard parallel.
But in some cases, for special reasons, an opposite course of procedure
has been followed, and subdivisional work has been begun on lh« north
boundary and has been extended southward and eastward or southward
"and westward.
In the more recent general instructions, greater care has been exer-
cised to secure rcctangujar subdivisions by fixing a strict limitation that
no new township exteriors or section lines shall depart from a true merid-
ian or east and west line more than twenty-one minutes of arc; and
that where a random line is found liable to correction beyond this limit,
a true line on a cardinal course must be run, setting a closing corner on
the line to which it closes.
This produces, in new surveys closing to irregular old work, a great
number of exteriors marked by a double set of corners. All retracing
surveyors should proceed i-nd^r these new conditions with full knowl-
edge of the field notes and exceptional methods of subdivision.
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73^ SURVEYING,
SYNOPSIS OF ACTS OF CONGRESS.
The first enactment in regard to the surveying of the public lanilj
was an ordinance passed by the Congress of the Confed-
the^^coSgwMi Sf eration May 20, 1785, prescribing the mode for the survey
timi orMfiS^*'^' °^ ^^^ ** Western Territory," and which provided that said
J7JJ5. u. s. Lan.l territory should be divided into ** townships of six miles
tiSn"i«BM?**' ^^^" square, by lines running due north and south, and others
crossing ihem at right angles " as near as might be.
It further provide^ that the first Tne running north and south should
begin on the Ohio River, at a point due north from the western terminus
of a line run as the south boundary of the State of Pennsylvania, and the
first line running east and west should begin at the same point and ex-
tend through the who'.e tcirircry. In these initial surveys only the ex-
terl«)r lines of the townships were surveyed, but the plats were marked by
subdivisions into sections 1 mile square, numbered from i to 36, com-
mencing with No. I in the southeast corner of the township, and run-
ning from south to north in each tier to No. 36 in the northwest comer
of the township; mile corners were establlshtd on the township lines.
The regir.n embraced by the surveys indcr this law forms a part of the
present Slate of Ohio, and is generally known as *' the Seven Ranges."
The Federal Congress passed a law, approved May 18, 1796, in regard
Act of May 18. ^^ Surveying the public domain, which applied to " the ter-
1796. u. 8. Stat- ritory northwest of the River Ohio, and above the mouth
;Sri.^4«6*5?S- of the Kentucky River."
SS^a^taYutwi Section 2 of said act provided for dividing such lands as
had not been already surveyed or disposed of " by north and
south lines run according to the true meridian, and by others crossing
them at right angles, so as to form townships of 6 miles square," etc.
It also provided thit " one-half cf said townships, taking them alter-
nately, should be subdivided into sections containing, as nearly as may
be, 640 acres each, by running through the same each way parallel lines
at the end of every two miles; and by marking a corner on each of said
lines at the end of every mile." The act also provided that " the sec-
tions shall be numbered, respectively, beginning with the number one
in the northeast section, and proceeding west and east alternately through
the township, with progressive numbers till the thirty-sixth be completed.**
This method of numbering sections is still in use.
An act amendatory of the foregoing, approved May 10, 1800, required
the '* townships west of th^ Muskingum, which are directed to be sold
Act of May 10 ^" quarter townships, to be subdivided into half sections of
1800. u. s. .Stat-' 320 acres each, as nearly as may be, by running parallel lines
?Jr2.p*73. SS- through the same from cast to west, and from south to north,
Reri^staYut^ ^* ^^^ distance of one mile from each other, and marking
corners, at the distance of each half mile on the lines ranning
from east to west, and at the distance of each mile on those running from
south to north. And the interior lines of townships intersected by the
Muskingum, and of all townships lying east of that river, which have not
been heretofore actually subdivided into sections, shall also be run and
marked * * *. And in all cases where the exterior lines of the townships
thus to be subdivided into sections or half sections shall exceed or shall
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APPENDIX /. 739
not extend six miles, the excess or deficiency shall be specially noted,
and added to or deducted from the western or northern ranges of sec-
tions or half sections in such townships, according as the error may be in
running the lines from east to west or from south to north." Said act
also provided that the northern and western tiers of sections should be
sold as containing only the quantity expressed on the plats, and all others
as containing the complete legal quantity.
The act approved June i, 1796, ** regulating the grants
of land appropriated for military services," etc., provided for Act of June 1,
dividing the "United States Military Tract," in the State ofSS S \iS^,
Ohio, into townships 5 miles square, each to be subdivided ^'<»i- 1» p- *•*•
into quarter town: hips cor.t:ining 4,000 a:rcs.
Section 6 of the act approved March i, 1800. amendatory of the fore-
going act, enacted that the Secretary of the Treasury was ^^^^j^^ .j^,
authorized to subdivide th€ quarter townships into lots of isoo. u. 8**8^1^
100 acres, bounded as nearly as practicable by parallel lines J^^g^^ j^'««'
160 perches in length by 100 perches in width These sub-
divisions into lots, however, were made upon the p'ats in the office oL the
Secretary of the Treasury, and the actual survey was only made at a sub-
sequent time when a sufficient number of such lots had been located to
warrant the survey. It thrs happened, in some instances, that when the
survey came to be made the plat and survey could not be made to agree,
and that fractional lots on plats were entirely crowded out. A knowl-
edge of this fact may explain some of the difficulties met with in the dis-
trict thus subdivided.
The act of Congress approved February 11, 1805, directs the subdivi-
sion of the public lands into quarter sections, and provides ActofFebmiuT
that all corners marked in the field shall be established as n.iwe. u.s.stat-
the proper corners of the sections or quarter sections which ?or2,jJ sisl^-
they were intended to designate, and that corners of half i{e?iB?d8tatut4!
and quarter sections not marked shall be placed as nearly as
possible " equidistant from those two corners which stand on the same
line." This act further provides that " the boundary lines actually run
and marked " (in the field) " shall be established as the proper boundary
lines of the sections, or subdivisions, for which they were intended, and
the length of such lines as returned by either of the surveyors aforesaid
shall be held and considered as the true length thereof. And the bound-
ary lines which shall not have been actually run and marked as afore-
said shall be ascertained by running straight lines from the established
corners to the opposite corresponding corners, but in those portions of
the fractional townships where no such opposite or corresponding cor-
ners have been or can be fixed, the said boundary lines shall be ascer-
tained by running from the established corners due north and south, or
east and west lines, as the case may be. to the watercourse, Indian bound-
ary line, or other external boundary of such fractional township."
The act of Congress approved April 24, 1820. provides for the sale of
public lands in half-quarter sections, and requires that "in .^„, * ..^
f ^, J. . . t ' X- . I !• t Aot Of April 94,
every case of the division of a quarter section the line for isao. u. s. stat-
the division thereof shall run north and south," " and frac- "0^3. p'sm^sSo-
tional sections, containing 160 acres and upwards, shall in '{j^taSistaSite'"
like manner, as nearly as practicable, be subdivided into half
quarter sections, under such rules and regulations as may be prescribed
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740 SURVEYING.
by the Secretary cf the Treasury; but fractional sections containing less
than i6o acres shall not be divided."
The act of Congress approved May 24, 1824. provides " that whenever.
Act of May »t, *" ^^^ Opinion of the President of the UniteJ States, a de-
1^. u. s- stot' parture from the ordinary mode of surveying land on any
?oL4,^s4V*^^' river, lake, bayou, or watercourse would promote the public
interest, he may direct the surveyor-general in whose district
such land is situated, and where the change is intended to be made, under
such rules and regulations as the President may prescribe, to cause the
lands thus situated to be surveyed in tracts of two acres in width, front-
ing on any river, bayou, lake, or watercourse and running back the
depth of forty acres."
The act of Congress approved April 5, 1832, directed the subdivision
Act of April ft ^^ ^^ public lands into quarter-quarter sections; that in
183?. u. s. Stat- every case of the division of a half-quarter section the divid-
^cJTi.pfBoa^^SoI^- ing line should run east and west, and that fractional sec-
ite?iiwiS^'ttttuui' tJons should be subdivided, under rules and regulations pre-
* scribed by the Secretary cf the Treasury. Under the latter
provision the Secretary directed that fractional sections containing less
than 160 acres, or the residuary portion of a fractional section, after the
subdivision into cs many quarter-quarter sections as it is susceptible o^
may be subdivided into lots, each containing the quantity of a quarter-
quarter section as nearly as practicable, by so laying down the line of
subdivision that they shall be 20 chains wide, which distances are to be
marked on the plat of subdivision, as are also the areas of the quarter
quarters and residuary fractions.
These two acts last mentioned provided that the corners and contents
of half-quarter and quarter-quarter sections should be ascertained as
nearly as possible in the manner and on the principles prescribed in the
act of Congress approved February 11, 1805.
GENERAL RULES.
From the foregoing synopsis of Congressional legislation it is evi-
dent—
I St. That the boundaries of the public lands established and returned
by the duly appointed Government surveyors, when approved by the
surveyors-general and accep cd by the Government, ere u c' anyablc.
2d. That tlie or"gi al t. wrs'ip. section, and quarter-section corners
established by the Government surveyors must stand as the true cor-
ners which they were intended to represent, whether the corners be in
place or not.
3d. That quarter-quarter corners not established by the Government
surveyors shall be placed on the straight Knes joining the section and
quarter-section corners and midway between them, except on the last
half mile of section lines c'osing on the north and west boundaries of
the township, or on otler Ins bclwecn fractional sections.
4th. Thr.t all subdivsional li-^es of a section running between corners
established in ihc orii^inal survey of a township must be straight lines,
running frrm the proper corner in ore ?e:tion line to its opposite ccr-
respondincf corner in the oppos'te section line.
5th. That in a fractional section where no opposite corresponding
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APPENDIX L 741
corner has been or can be established, any required subdivision line of
such section must be run from the proper original corner in the boundary
line due east and west, or north and south, as the case may be, to the
watercourse, Indian reservation, or other boundary of such section, with
due parallelism to section lines.
From the foregoing it will be plain that extinct corners of the Gov-
ernment surveys must be restored to their original locations, whenever
it is possible to do so; and hence resort should always be first had to
the marks of the survey in the field. The locus of the missing corner
should be first identified on the ground by the aid of the mound, pits,
line trees, bearing trees, etc., described in the field notes of the original
survey.
The identification of mounds, pits, wilness trees, or other permanent
objects noted in the field noles of survey, affords the best means of re-
locating the missing corner in its original position. If this cannot be
done, clear and convincing testimony of citizens as to the locality it
originally occupied should be taken, if such can be obtained. In any
event, whether the locus of the corner be fixed by one means or the
other, such locus should always be tested and confirmed by measure-
ments to ktiozvn corners. No definite rule can be laid down as to what
shall be sufficient evidence in such cases, and much must be left to the
skill, fidelity, and good judgment of the surveyor in the performance of
his work.
EXCEPTIONAL CASES.
When new measurements are made on a single line to determine the
position thereon for a restored lost corner (for example, a quarter-section
corner on line between two original section corners), or when new meas-
urements are made between original corners on two lines for the pur-
pose of fixing by their intersection the position of a restored missing
corner (for example, a corner common to four sections or four town-
ships), it will almost invariably happen that discrepancies will be devel-
oped between the new measurements and the original measurements in
the field notes. When these differences occur the surveyor will in all
cases establish the missing corner by proportionate measurements on
lines conforming to the original field notes and by the method followed
in the original survey. From this rule there can be no departure, since
it is the basis upon which the whole operation depends for accuracy and
truth.
In cases where the relocated corner cannot be made to harmonize
with the field notes in all directions, and unexplained error in the first
survey is apparent, it sometimes becomes the task of the surveyor to
place it according to the requirements of one line and against the calls
of another line. For instance, if the line between sections 30 and 31,
reported 78 chains long, would draw the missing corner on range line
I chain eastward out of range with the other exterior corners, the pre-
sumption would be strong that the range line had been run straight
and the length of the section line wrongly reported, because experience
shows that west random lines are regarded as less important than range
lines and more liable to error.
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74^ SURVEYING.
Again, where a corner on a standard parallel has been obliterated, it
is proper to assume that it was placed in line with other corners, and
if an anomalous length of line reported between sections 3 and 4 would
throw the closing corner into the northern township, a surveyor would
properly assume that the older survey of the standard line is to control
the length of the later and minor line. The marks or corners found on
such a line closing to a standard parallel fix its location, but its length
should be limited by its actual intersection, at which point the lost closing
corner may be placed.
The strict rule of the law that ** all comers marked in the field shall
be established as the corners which they were intended to designate,"
and the further rule that " the length of lines returned by the surveyors
shall be held and considered as the true length thereof," are found in
some cases to be impossible of fulfilment in all directions at once, and a
surveyor is obliged to choose, in his own discretion, which of two or
more lines must yield, in order to permit the rules to be applied at all.
In a case of an erroneous but existing closing comer, which was set
some distance out of the tme State boundary of Missouri and Kansas,
it was held by this office that a surveyor subdividing the fractional sec-
tion should preserve the boundary as a straight line, and should not re-
gard said closing corner as the proper corner of the adjacent fractional
lots. The said corner was considered as fixing the position of the line
between two fractional sections, but that its length extended to a new
comer to be set on the true boundary line. The surveyor should there-
fore preserve such an original corner as evidence of the line; but its
erroneous position cannot be allowed to cause a crook between mile
corners of the original State boundary.
It is only in cases where it is manifestly impossible to carry out the
literal terms of the law, that a surveyor can be justified in making such a
decision.
The principle of the preponderance of one line over another of less
importance has been recognized in the rule for restoring a section corner
common to two townships in former editions of this circular. The new
corner should be placed on the township line; and measurements to
check its position by distances to corners within the townships are useful
to confirm it if found to agree well, but should not cause it to be placed
off the line if found not to agree, if the general condition of the boundary
supports the presumption that it was properly alined.
TO RESTORE LOST OR OBLITERATED CORNERS.
I. To restore corners on base lines and standard parallels. — Lost or
obliterated standard corners will be restored to their original positions
on a base line, standard parallel, or correction line, by proportionate
measurements on the line, conforming as nearly as practicable to the
original field notes and joining the nearest identified original standard
corners on opposite sides of the missing corner or corners, as the case
may be.
(0) The term " standard corners " will be understood to designate
standard township, section, quarter section, and meander corners; and,
in addition, closing corners, as follows: Closing corners used in the
original survey to determine the position of a standard parallel, or cs-
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APPENDIX I. 743
tablished during the survey of the same, will, with the standard corners,
govern the alinement and measurements made to restore lost or oblit-
erated standard corners; brt no other closing corners will control in
any manner the restoration of standard corners on a base lin-e or standard
parallel.
{b) A lost or obliterated closing corner from which a standard parallel
has been initiated or to which it has been directed will be reestablished
in its original place by proportionate measurement from the corners used
in the original survey to determine its position. Measurements from
corners on the opposite side of the parallel will not control in any manner
the relocation of said corner.
{c) A missing closing corner originally established during the survey
of a standard parallel as a corner from which to project surveys south
will be restored to its original position by considering it a standard cor-
ner and treating it accordingly.
(d) Therefore, paying attention to the preceding explanations, we
have for the restoration of one or several corners on a standard parallel,
and for general application to all other surveyed lines, the following pro-
portion:
As the original field-note distance between the selected known corners
is to the new measure of said distance, so is the original field-note length
of any part of the line to the required new measure thereof.
The sum of the computed lengths of the several parts of a line must
be equal to the new measure of the whole distance.
(e) As has been observed, existing original corners cannot be dis-
turbed; consequently, discrepancies between the new and the original
field-note measurements of the line joining the selected original corners
will not in any manner affect measurements beyond said corners, but
the differences will be distributed proportionately to the several intervals
embraced in the line in question.
(0 After having checked each new location by measurement to the
nearest known corners, new corners will be established permanently and
new bearings and measurements taken to prominent objects, which should
be of as permanent a character as possible, and the same recorded for
future reference.
2. Restoration of township corners common to four townships. — Two cases
should be clearly recognized: ist. Where the position of the original
township corner has been made to depend upon measurements on two
lines at right angles to each other. 2d. Where the original corner has
been located by measurements on one line only; for example, on a guide
meridian.
(a) For restoration of a township corner originally subject to the
first condition: A line will first be run connecting the nearest identified
original corners on the meridional township lines, north and south of
the missing corner, and a temporary corner will be placed at the proper
proportionate distance. This will determine the corner in a north and
south direction only.
Next, the nearest original corners on the latitudinal township lines
will be connected and a point thereon will be determined in a similar
manner, independent of the temporary corner on the meridional line.
Then through the first temporary corner run a line east (or west) and
through the second temporary corner a line north (or south), as relative
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J^44 BVkVEViNG.
situations may suggest. The intersection of the two lines last run will
define the position of the restored township corner, which may be per-
manently established.
(6) The restoration of a lost or obliterated township corner estab-
lished under the second conditions, i.e., by measurements, on a single
line, will be effected by proportionate measurements on said line, between
the nearest identified origmal corners on opposite sides of the missing
township corner, as before described.
3. Recstablishment of corners common to two townships. — The two nearest
known corners on the township line, the same not being a base or a cor-
rection line, will be connected as in case No. i, by a right line, and the
missing corner established by proportionate distance as directed in that
case; the location thus found will be checked upon by measurements to
nearest known section or quarter-section corners north and south, or
east and west, of the township line, as the case may be,
4. ReestablishmcHt of closing corners. — Measure from the quarter-sec-
tion, section, or township corner east or west, as the case may be, to the
next preceding or succeeding corner in the order of original establishment,
and reestablish the missing closing corner by proportionate measurement.
The line upon which the closing corner was originally established should
always be remeasured, in order to check upon the correctness of the new
location. See page 648 flF. for details.
5. Reestahlishment of interior section corners. — This class of comers
should be reestablished in the same manner as corners common to four
townships. In such cases, when a number of comers are missing on all
>ides of the one sought to be reestablished, the entire distance must, of
:ourse, be remeasured between the nearest existing recognized comers
both north and south, and east and west, in accordance with the rule
laid down, and the new corner reestablished by proportionate measure-
ment. The mere measurement in any one of the required directions will
not suffice, since the direction of the several section lines running north-
ward through a township, or running east and west, are only in the
most exceptional cases true prolongations of the alinement of the sec-
tion lines initiated on the south boundary of the township; while the
east and west lines running through the township, and theoretically sup-
posed to be at right angles with the former, are seldom in that condi-
tion, and the alinements of the closing lines on the east and west bound-
aries of the township, in connection with the interior section lines, are even
less often in accord. Moreover, the alinement of the section line itself
, from corner to corner, in point of fact, also very frequently diverges
from a right line, although presumed to be such from the record con-
tained in the field notes and so designated on the plats, and becomes
either a broken or a curved line. This fact will be determined, in a
timbered country, by the blazes which may be found upon trees on either
side of the line, and although such blazed line will not strictly govern
as to the absolute direction assumed by such line, it will assist very
materially in determining its approximate direction, and should never
be neglected in retracements for the reestahlishment of lost corners of
any description. Sight trees described in the field notes, together with
the recorded distances to same, when fully identified, will, it has been
held, in one or more States, govern the line itself, even when not in a
direct or straight line between established corners, which line is then
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AAF^A'Jl/^:'/. 745
Wjf^axily.A. brOl^fv Unt by passing through s^id gigbt-tree?. Such
tefitni^ic^^.in. C«i3t,encc and properly identified beyond a question ot
i$>ntei; wiil .very, materially assist in evidencing the correct relocation of
a missing corner. It is greatly to be regretted that the earlier field notes
of. purvey are so very meagre in the notation of the topography found
on the, original line, which might in very many instances materially lessen
si 5uryeyor'5 labors in retracement of lines and reestablishment of the
required missing corner. In the absence of such sight trees and othef
eyidencp regarding the line, as in an open country, or where such evi-
dence Thas been destroyed by time, the elements, or the progress of im-
provement, the line connecting the. known comers should be run straight
from cprner. to cprner.
6-, RefstablUhptent -^f quarl^r-sedkn corner^ en township boundaries. -r
Only.on^ set of c|uarie.r-sectian corners are actually marked in the field
on. township lines, and they afe established at the time when the town-
ship /exteriors are run. When double section corners are found, the
qiU^rt^r-seotion corners are considered g^r^erally as standing midway
b^tWiCen the corners of their respective sectionsj and when required to
be established or .reestablished, as the case may. be, they should be gen-
erally ?o placed; but gteat care should be exercised not to mistake the
corners belonging, to one. township for those of another. After deter-
mining thfe proper section corners marking the Jine upon which the
missing quarter-section corner is to be reestablished, and -measuring
said Une, the jnissing q.uarter-section corner will be reestablished in aCr
cordance with the requirements of the original field notes of survey, by
propoi^ipUate isneaslirj^ment %belween the . section corners marking the
Une^-.-».- .'.... ■.•.■..:■•.. 5 • . ..•.:;:
'. .Where there are double sets.of section corners on township and range
jtines, t and - the quarteT-sectiptf corners for sections south of the town^
^hip/Or east of the. range lines; are required to be established in. the field,
the said quarter-section corners should be so placed as to suit the cal^
Cttlatipn'of arctas of the quarter sections adjoining the township boundr
"aries as expressed upon the official township plat, adopting proportionate
mea^rem^nts when, the present measurement cf the north and west
.boundaries of the .sectidn differ from the original measurements.
' 7v Reestablishment of qiMrter-section corners on closing section lines ber
Hveen fractional sections, -r-This. class o£ corners must be reesta'blished ac-
cording to the original measurement of 40 chains from the last interior
^fttion cotner; - If ^ the mtasnremertts do nat agree with the original
survey, the excess or deficiency must be divided proportionately be-
tween the two distances, as expre^ssed in the field notes of original suc-
•yey. The section corner started from and the corner closed upon should
be: connected by a right lirie, unless the retracement should develop the
fact. that the section line is either a broken or curved line, as is some-
times thef case.'
8; Reestablishment of interior quarter-section corners: — In some of the
^Ider surveys these corners are placed at variable distances, in which
,caSe the field notes, of the original survey must be consulted, and the
-quarter-section corner reestablished at proportionate distances between
the corresponding section corners, in accordance therewith. The later
.^surveys being more uniform and in stricter accordajice with law, Uhe
.pijssing quatter-section comer. must, be .reestablisjjed equidista^^t. between
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746 SURVEYING.
the section corners marking the line, according to the field notes of the
original survey. The remarks made under section 5, in relation to section
lines, apply with full force here also; the caution there given not to
neglect sight trees is equally applicable, since the proper reestablishment
of the quarter-section comer may in some instances very largely depend
upon its observance, and avoid one of the many sources of litigation.
9. Where double comers were originally established, one of which is stand-
ing, to reestablish the other,— It being remembered that the comers es-
tablished when the exterior township lines were run belong to the sec-
tions in the townships north and west of those lines, the surveyor must
first determine beyond a doubt to which sections the exising corner be-
longs. This may be done by testing the courses and distances to wit-
ness trees or other objects noted in the original field notes of survey,
and by remeasuring distances to known comers. Having determined
to which township the existing comer belongs, the missing comer may
be reestablished in line north or south of the existing comer, as the
case may be, at the distance stated in the field notes of the original sur-
vey, by proportionate measurement, and tested by retracement to the
opposite corresponding comer of the section to which the missing sec-
tion corner belongs. These double comers being generally not more
than a few chains apart, the distance between them can be more ac-
curately laid off, and it is considered preferable to first establish the miss-
ing corner as above, and check upon the corresponding interior comer,
than to reverse the proceeding; since the result obtained is every way
more accurate and satisfactory.
10. Where double comers were originally established, ard both are miss-
ing, to reestablish the ofie established when the township line was run. — The
surveyor will connect the nearest known corners on the township line
by a right line, being careful to distinguish the section from the closing
corners, and reestablish the missing comer at the point indicated by the
field notes of the original survey by proportionate measurement. The
corner thus restored will be common to two sections either north or west
of the township boundary, and the section north or west, as the case may
be, should be carefully retraced, thus checking upon the reestablished cor-
ner, and testing the accuracy of the result It cannot be too much
impressed upon the surveyor that any measurements to objects on line
noted in the original survey are means of determining and testing the
correctness of the operation.
11. Where double comers were originally established, and bcth are miss-
ing, to reestablish the one established when the township wcu subdivided. — ^The
comer to be reestablished being common to two sections south or east
of the township line, the section line closing on the missing section comer
should be first retraced to an intersection with the township line in the
manner previously indicated, and a temporary comer established at the
point of intersection. The township line will of course have been pre-
viously carefully retraced in accordance with the requirements of the
original field notes of survey, and marked in such a manner as to be
readily identified when reaching the same with the retraced section line.
The location of the temporary corner planted at the point of intersection
will then be carefully tested and verified by remeasurements to objects
and known comers on the township line, as noted in the original field
notes of survey, and the necessary corrections made in such relocation.
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APPENDIX I. '747
A permanent corner will then be erected at the corrected location on the
township line, properly marked and witnessed, and recorded for future
requirements.
12. Where triple corners were originally established on range lines, one
or two of which have become obliterated, to reestablish either of them. — it
will be borne in mind that only two corners were established as actual
comers of sections, those established on the range line not correspond-
ing with the subdivisional survey east or west of said range line. The
surveyor will, therefore, first proceed to identify the existing comer or
corners, as the case may be, and then reestablish the missing comer
or comers in line north or south, according to the distances stated in
the original field notes of survey in the manner indicated for the re-
establishment of double comers, testing the accuracy of the result 'ob-
tained, as hereinbefore directed in other cases. If, however, the dis-
tances between the triple corners are not stated in the original field
notes of survey, as is frequently the case in the returns of older surveys,
the range line should be first carefully retraced, and marked in a man-
ner sufficiently clear to admit of easy identification upon reaching same
during the subsequent proceedings. The section lines closing upon the
missing corners must then be retraced in accordance with the original
field notes of survey, in the manner previously indicated and directed,
and the comers reestablished in the manner directed in the case of double
corners. The Surveyor cannot be too careful, in the matter of retrace-
ment, in following closely all the recorded indications of the original
line, and nothing, however slight, should be neglected to insure the
correctness of the retracement of the original line; since there is no
other check upon the accuracy of the reestablishment of the missing
comers, unless the entire corresponding section lines are remeasured by
proportional measurement and the result checked by a recalculation of
the areas as originally returned, which, at best, is but a very poor check,
because the areas expressed upon the margin of many plats of the older
surveys are erroneously stated on the face of the plats, or have been care-
lessly calculated.
13. Where triple comers were originally established on range lines, aU of
which are missing, to reestablish same. — These corners should be reestab-
lished in accordance with the foregoing directions, commencing with the
corner originally established when the range line was run, establishing
the same in accordance with previously given directions for restoring sec-
tion and quarter-section comers; that is to say, by remeasuring between
the nearest known corners on said township line, and reestablishing the
same by proportionate measurement. The two remaining will then be
reestablished in conformity with the general rules for reestablishment of
double corners.
14. Reestablishment of meander comers. — Before proceeding with the
reestablishment of missing meander corners, the surveyor should have
carefully rechained at least three of the section lines between known
comers of the township within which the lost corner is to be relocated,
in order to establish the proportionate measurement to be used. This
requirement of preliminary remeasurement of section lines must in no
case be omitted; since it gives the only data upon which the fractional
section line can be remeasured proportionately, the comer marking the
terminus, or the meander comer, being missing, which it is intended to
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748
SUJ^VEYIN'G,
reestablish. The missing meander corner will be reestablished on the
section or township line retraced in its original location, by the propor-
tionate measurement found by the preceding operations, from the nearest
known corner on such township or section line, in accordance with the
requirements of the original field notes of survey.
Meander corners hold the peculiar position of denoting a point on
line between landowners, without usually being the legal terminus or
corner of the lands owned. Leading judicial decisions have affirmed
that meander lines are not strictly boundaries, and do not limit the
ownership to the exact areas placed on the tracts, but that said title ex-
tends to the water, which, by the plat, appears to bound the land.
As such water boundaries are, therefore, subject to change by the en-
croachment or recession of the stream or lake, the precise location of old
meanders is seldom important, unless in States whose laws prescribe that
dried lake beds are the property of the State.
Where the United States has disposed of the fractional lots adjacent
to shores, it claims no marginal lands left by recession or found by rea-
son of erroneous survey. The lines between landowners are therefore
regarded as extended beyond the original meander line of the shore, but
the preservation or relocation of the meander corner is important, as
evidence of the position of the section line.
The different rules by which division lines should be run between pri-
vate owners of riparian accretions are a matler of State legislation, and
not subject to a general rule of this oftice.
15. Fractional section lines.— Conniy and local surveyors being some-
times called upon to restore fractional section lines closing upon Indian,
military, or oiher reservations, private p rants, etc., such lines should be
restored upon the same principles as directed in the foregoing pages, and
checked whenever possible upon such corners or monuments as have
been placed to mark ^uch boundary lines.
In some instances corners have been moved from their original posi-
tion, either by accident or design, and county surveyors are called upon
to restore such corners to their original positions, but, owing to the
absence of any and all means of identification of such location, are un-
able to make the result of their work acceptable to the owners of the
lands aflfected by such corner. In such cases the advice of this office has
invariably been to the effect that the relocation of such corner must be
made in accordance with the orders of a court of competent jurisdiction,
the United States having no longer any authority to order any changes
where the lands affected by such corner have been disposed of.
RECORDS.
The original evidences of the public-land surveys in the following
States have been transferred, under the provisions of sections 2218, 2219.
and 2220, United States Revised Statutes, to the State authorities, to
whom application should be made for such copies of the original plats
and field notes as may be desired, viz.:
Alabama: Secretary of State. Tilontgomery.
Arkansas: Commissioner of State Lands, Little Rock.
Illinois: Auditor of State. Springfield.
Indiana: Auditor of State, Indianapolis.
Iowa: Secretary of State, Des Moines.
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APPENDIX L 749
Kansas: Auditor of State and Register of State Lands, Topeka.
Michigan: Commissioner of State Land Office, Lansing.
Mississippi: Commissioner of State Lands, Jackson.
Missouri: Secretary of State, Jefferson City.
Nebraska: Commissioner of Public Lands and Buildings, Lincoln.
Ohio: Auditor of State, Columbus.
Wisconsin: Commissioners of Public Lands, Madison.
In other public-land States the original field notes and plats are re-
tained in the offices of the United States surveyors-general.
SUBDIVISION OF SECTIONS
This office being in receipt of many letters making inquiry in regard
to the proper method of subdividing sections of the public lands, the fol-
lowing general rules have been prepared as a reply to such inquiries. The
rules for subdivision are based upon the laws governing the survey of
the public lands. When cases arise which are not covered by these rules,
and the advice of this office in the matter is desired, the letter of inquiry
should, in every instance, contain a description of the particular tract or
corner, with reference to township, range, and section of the public sur-
veys, to enable the office to consult the record; also a diagram showing
conditions found:
1. Subdivision of sections into quarter sections. — Under the provisions
of the act of Congress approved February ii, 1805, the course to be pur-
sued in the subdivision of sections into quarter sections is to run straight
lines from the established quarter- section corners. United States surveys,
to the opposite corresponding corners. The point of intersection of the
lines thus run will be the corner common to the several quarter sections,
or, in olher words, the legal centre of the section.
(a) Upon the lines closing on the north and west boundaries of a
township, the quarter-section corners are established by the United State
deputy surveyors at 40 chains to the north or west of the last interior
section corners, and the excess or deficiency in the measurement is thrown
into the half mile next to the township or range line, as the case may be.
(b) Where there are double sets of section corners on township and
range lines, the quarter corners for the sections south of the township
lines and east of the range lines are not established in the field by the
United States deputy surveyors, but in subdividing such sections said
quarter corners should be so placed as to suit the calculations of the areas
of the quarter-sections adjoining the township boundaries as expressed
upon the official plat, adopting proportionate measurements where the
new measurements of the north or west boundaries of the section differ
from the original measurements.
2. Subdivision of fractional sections. — Where opposite corresponding
corners have not been or cannot be fixed, the subdivision lines should be
ascertained by running from the established corners due north, south,
*!ast, or west lines, as the case may be, to the watercourse, Indian bound-
ary line, or other boundary of such fractional section.
(a) The law presumes the section lines surveyed and marked in the
field by the United States deputy surveyors to be due north and south
or east and west lines, but in actual experience this is not always the
case. Hence, in order to carry out the spirit of the law, it will be nec-
essary in running the subdivisional line through fractional sections to
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7SO SURVEYING.
adopt mean courses where the section lines are not due lines, or to run
the subdivision line parallel to the east, south, west, or north boundary
of the section, as conditions may require, where there is no opposite
section line.
3. Subdivision of quarter sections into quarter quarters. — Preliminary to
the subdivision of quarter sections, the quarter-quarter comers will be
established at points midway between the second and quarter-section
comers, and between quarter corners and the centre of the section, ex-
cept on the last half mile of the lines closing on the north or west bound-
aries of a township, where they should be placed at 20 chains, propor-
tionate measurement, to the north or west of the quarter-section comer.
(a) The quarter-quarter section corners having been established as
directed above, the subdivision lines of the quarter section will be run
straight between oposite corresponding quarter-quarter section comers
on the quarter-section boundaries. The intersection of the lines thus run
will determine the place for the corner common to the four quarter-
quarter sections.
4. Subdivision of fractional quarter sections. — ^The subdivision lines of
fractional quarter sections will be mn from properly established quarter-
quarter section comers (paragraph 3) due north, south, cast, or west,
to the lake, watercourse, or reservation which renders such tracts frac-
tional, or parallel to the east, south, west, or north boundary of the
quarter section, as conditions may require. (See paragraph 2 (a).)
5. Proportionate measurement. — By "proportionate measurement," as
used in this circular, is meant a measurement having the same ratio to
that recorded in the original field notes as the length of chain used in the
new measurement has to the length of chain used in the original sur-
vey, assuming that the original and new measurements have been cor-
rectly made.
For example: The length of the line from the quarter-section comer
on the west side of sec. 2, T. 24 N., R. 14 E., Wisconsin, to the north
line of the township, by the United States deputy surveyor's chain, was
reported as 45.40 chains, and by the county surveyor's measure is re-
ported as 4290 chains; then the distance which the quarter-quarter section
comer should be located north of the quarter-section comer would be
determined as follows:
As 45.40 chains, the Government measure of the whole distance, is to
42.90 chains, the county surveyor's measure of the same distance, so is
20.00 chains, original measurement, to 18.90 chains by the coun^ sur-
veyor's measure, showing that by proportionate measurement in this case
the quarter-quarter section comer should be set at 18.90 chains north of
the quarter-section corner, instead of 20.00 chains north of such corner,
as represented on the official plat. In this manner the discrepancies be-
tween original and new measurements are equitably distributed.
S. W. Laicoreux,
Commissiamer.
Department of the Interior,
October 16, 1896.
Approved:
David R. Francis,
Secretary,
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TABLES.
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TABLE I.
Trigonobietric Formula
TRiooNomTRio Fun cnoMs.
Let A (Fig. 107) = angle BACss arc BF^ and let the radius AJT » AB ==
AH=1.
We then have
sin^
^BO
cos^
= AO
tan^
^DF
cot^
= HQ
0ee^
= AJ)
ooeeo A = AO
▼erain A = CF = BE
covers X = BK = HL
exsec ^ = BD
coezsec ^ = BG
chord ^ = Bi**
chord 2 A = BI^ 2BC
In the right-angled triangle ABC (Fig, 107)
Let AB = c,AC^ 6, and BC m a.
We then have :
Fio. 107,
1. sin^
2. COS A
8. tan^
4. cot^
5. sec^
= — = cosB
c
b
= — = sin B
c
a
= -jT- =cotJ^
b
= — = tan if
a
c
= -i- =coseoB
6. coieo^ = — =8ecif
a
m A C — 6 _
7. vers -4 = — - — =5 covers if
c
c -6
= ~i, - = coexaec B
9. covers -4 = ^-^~ = yersin B
a ezseoX
10. coexBeo^ =
c
c —a
=: ezsecB
SI. area =
11. a =csin>4 s=5tan^
12. 6 =:cco6^ = acot^
13. C = -; 7 =
sin ^ cos^
14. a = ccosJ^s frcotB
15. & = o sin B = a tan B
16. c = **- = - ^-.
cos B sin B
17. a = 'i/Jj^hfi^r^
19. e = Va«"-H>«
20. C?s=90* = ^-f-B
3
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754
SURVEYING
TABLE l.'^CofUinued.
Trigonombtrig Formuub.
Solution ov Obu^ub TBuvoLn.
Fio. 10&
90
27
S8
29
80
A,B,a
A,a,h
C,a.6
a,5,«
as
''tf^.Aa
C,6,c
HU-B)
A,B
^=iiH^— ^'
a •
iC.
J9 = HU + B)-HU--B)
C=«(a + 6)'^fT7^
cosHU + B),
a)(a-6)(»~c)
6 c '
cos il = -—
a< sin B . sin C
2 8in^
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TABLES,
755
TABLE L^GrnHnuid.
TeIGONOMETRIC FORMULiB.
OmnUL fOBMULM.
^^ = Vl-co-^ = tan^oc^
OOS^ r>
1
= V 1 - stns ^
oot^itn^
OOS^ ss
tanwl SK
tanwl »
cotA o
BOO A
1
sin ^
/■■
OO0*^
V flec« -4—1
V 1 — COS* ^
COB^
Bin 2^
l+oos2^
■inSwl
Terg2^
8in2^
=s ezaeowlcotH^
1
tana
Btn ^
_8ln24_
1 — cobS^
OOt^ as ^-^"LT^^
cot ^ =9 -■
46
▼ers^
47
venA
48
exaeoA
48
BinH^
10
Bin 2^
U
cobH^
ft2
O08 2^
w ^ coBec* ^ — 1
Bin2X ac ^+<^o»^'^
verB2^ °° sin 2^
1 — OOB^ s Bin^tanH^ « 9ttak^^A
KoeoAeoBA
tBnHA
A
» aec^ — 1 s tanwitAiiH^
"cob -4
y 2 - y a
s 28ill^008^
■/
14-ooa4
2
Digitized by
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75^ SVUVMYWa.
TABLE l.-^Continued.
Trigonometric FoRMULiS.
1
QmSRAL FOBMULA.
-.- ^ ... tan-4 ^ ^ . 1 — C08-4 ,/i — oos^
Stan A
*«.
l-tan«^
66.
oot.H^«
1
ocweo
.^— oot^
66. oot 8 X s
Sootwl
67.
▼eP8Hii =
%yer%A
1-
-COS -4
6&Ten8^=8slii<^
1-— oos^
60. ezsecH^s
60. ezsec % A ^
(1 + 008-4) + f^8(l H- cos ui)
tan«X
1 — tan* A
61. sin (ii ± B) = Bin j4 . C08 B ± sin B . cos ^
02. cos (.4 ± B) = cos A . cos B 7 sin .4. sin J9
68. 8in^H-8inB = 88inHUH-B)cos^U~^
04. 8in^'-8lnBa8cosHM4-B)8inMU — ^
06. cos^ + cosBsScosHU + B)cosMU-B)
66. cosB — cos^ = 88inH(^H-B)8in^(2l — B)
07. sins^~8tnSB = coBSB — cos>^ = sinU + B)8inU— B^
Oa cos*^-8lnSB = oo8U+B)co8(^ — B)
fl». tan^ + tanB=-"*"i^ + ^
' cos A . cos B
70. Ua^-UnB = -^^44^I^,
cos .4. cos B
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I, Laprithit of Mwfcers, IL Ugirlttile Sins, ite., lar mrf TiA^
No.
lO
II
12
13
14
15
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17
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31
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56
57
58
59
60
liOff.
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041
079
114
146
176
204
230
255
279
301
322
342
362
380
398
415
431
447
462
477
491
505
519
531
544
556
568
580
591
602
613
623
633
643
653
663
672
681
690
699
708
716
724
732
740
748
756
763
771
778
41
38
35
32
30
28
26
25
24
22
21
20
20
18
18
17
16
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15
14
14
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12
12
12
II
II
II
10
10
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10
10
9
9
9
9
9
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8
8
8
8
8
7
8
7
No.
eo
61
62
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100
lOI
102
103
104
105
106
107
.108
109
110
Liosr.
778
785
792
799
806
813
820
826
833
839
845
851
857
863
869
875
881
886
892
898
903
908
914
919
924
929
934
940
944
949
954
959
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968
973
978
982
987
991
996
000
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009
013
017
021
025
029
033
037
041
ABO
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
I.I
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
3.1
2.2
2.3
3.4
2.5
2.6
2.7
2.8
2.9
3.0
31
3.2
3-3
3-4
3.5
3.6
3-7
3.8
3.9
4.0
4.1
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5.0
ARC
SIN
7.242
7-543
7.719
7.844
7-941
8.020
8.087
8.145
8.196
8.242
8.283
8.321
8.356
8.388
8.418
8.446
8.472
8-497
8.521
8.543
8.564
8.584
8.603
8.623
8.640
8.657
8.673
8.689
8.704
8.719
8.733
8.747
8.760
8.773
8.786
8.798
8.810
8.821
8.833
8.844
8.854
8.865
8.875
8.885
8.895
8.904
8.913
8.923
8.932
8.940
COS
JMO:
301
176
125
97
79
67
58
51
46
41
38
35
32
30
28
36
25
24
22
21
30
19
19
18
17
16
16
15
15
14
14
13
13
13
13
13
II
13
II
10
10
II
10
10
9
9
10
9
8
Diir.
TAN
7.242
7.543
7.719
7.844
7.941
8.020
8.087
8.145
8.196
8.242
8.283
8.321
8.356
8.388
8.418
8.446
8.472
8.497
8.521
8.543
8.564
8.585
8.604
8.623
8.640
8.657
8.674
8.689
8.705
8.719
8.734
8.747
8.761
8.774
8.786
8.799
8.811
S.823
8.834
8.845
8.855
8.866
8.876
8.886
8.896
8.906
8,915
8.934
8.933
8.943
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r
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89.9
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301
176
las
96
79
67
58
50
46
41
37
34
3a
29
a7
26
90*
89
~
8.24a
8.242
1.758
0.000
89.8
a
8.543
8.543
30X
176
ia6
97
80
1.457
0.000
88
89.7
3
8.719
8.719
1. 381
9.999
87
89.6
4
8.844
8.845
1. 155
9.999
86
89.5
5
8.940
8.94a
1.058
9-998
85
89.4
6
9.019
9.02a
67
59
5a
46
43
38
36
34
31
39
38
a7
a5
24
23
33
32
31
30
19
19
19
18
17
18
17
17
16
0.978
9-998
84
89.3
7
9.086
9.089
0.911
9-997
83
89.2
8
9.144
9.148
0.853
9.996
83
89.1
89.0
88.9
9
10
II
9.194
9.200
0.800
9.995
81
80
79
9.240
9.246
0.754
9-993
9.281
9.289
0.711
9.99a
88.8
la
9.318
9-3a7
0.673
9.990
78
88.7
13
9-35a
9.363
0.637
9.989
77
88.6
14
9.384
9.397
0.603
9.987
76
88.5
15
9.413
9.4a8
0.57a
9-985
75
88.4
16
9.440
9-457
0.543
9.983
74
88.3
17
9.466
24
23
ai
ao
ao
18
17
17
16
15
15
14
13
13
12
9.485
0.515
9.981
73
88.3
18
9.490
9.51a
0.488
9.978
7a
88.1
88.0:
87.9
19
20
ai
9-513
9.537
0.463
9.976
71
70
69
9-534
9.561
1 0.439
9.973
9-554
9.584
0.416
9.970
87.8
aa
9-574
9.606
0.394
9.967
68
87.7
a3
9.59a
9.6a8
0.37a
9.964
67
87.6
a4
9.609
9.649
0.35X
9.961
66
87.5
as
9.626
9.669
0.331
9.957
65
87.4
a6
9.64a
9.688
0.313
9.954
64
87.3
a7
9.657
9-707
0.393
9.950
63
87. a
a8
9.67a
9.726
0.374
9.946
6a
87.x
87.0
86.9
aq
30
31
9.686
9-744
0.256
9.94a
61
60
59
9.699
9.761
0.239
9-938
9.71a
9.779
0.33I
9-933
86.8
3a
9.724
9-796
0.304
9.928
58
86.7
33
9- 736
1«
12
9.813
0.187
9.924
57
86.6
34
9.748
9.839
16
O.I7I
9.919
56
86.5
35
9-759
Tf>
9.845
16
0.155
9-913
55
86.4
36
9.769
10
9.861
16
0.139
9-908
54
86.3
37
9.779
10
9.877
16
0.133
9.90a
53
86. a
38
9.789
10
9
9
9
Q
9.893
15
16
15
15
16
0.107
9.897
5a
86.1
86.0
85.9
39
40
41
9.799
9.908
0.093
9.891
51
50
49
9.808
9.9a4
0.076
9.884
9.817
9-939
0.061
9.878
85.8
4a
9.826
9-954
0.046
9.871
48
85.7
43
9.834
0
8
7
9.970
15
15
0.030
9.864
47
85.6
44
9.84a
9.985
0.015
9.857
46
85.5
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0.000
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og. i.758»«
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•.55630
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* Min. m lA
if .747 1
og. S.3«443
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44 u
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S.11961
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TABLES,
717
TABLE II.
For Converting Metres, Feet, and Chains.
Mbtrbs to Fekt.
Fret
TO Metres and
Chains.
Chains
TO Feet.
Metres.
Feet.
Feet.
Metres.
Chains.
Chains.
•
Feet.
I
3.28087
I
0.304797
0.0151
0.01
0.66
2
6.56174
2
0.609595
.0303
.02
1.32
3
9.84261
3
0.914392
.0455
.03
1.98
4
13.12348
4
I.219189
.0606
.04
2.64
5
16.40435
5
1.523986
.0758
.05
3 30
6
19.68522
6
1.828784
.0909
.06
3.96
7
22 . 96609
7
2.133581
.lo6l
.07
4.62
8
26.24695
8
2.438378
.1212
.08
5.28
9
29.52782
9
2.743175
.1364
.09
5.94
lO
32.80869
10
3.047973
.1515
.10
6.60
20
65.61739
20
6.095946
.3030
20
13-20
30
98.42609
30
9.143918
.4545
.30
19.80
40
131.2348
40
12.19189
.6061
.40
26.40
50
164.0435
50
15.23986
.7576
.50
33.00
60
196.8522
60
18.28784
.9091
.60
39.60
70
229.6609
70
21.33581
1.0606
•70
46.20
80
262.4695
80
24.38378
I.2I2I
.80
52.80
90
295.2782
90
27.43175
1.3636
.90
59.40
IOC
328.0869
100
30.47973
I.515I
I
66.00
200
656.1739
TOO
60.95946
3.0303
2
132
300
984 . 2609
300
91.43918
4-5455
3
198
400
1312.348
400
121.9189
6.0606
4
264
500
1640.435
500
152.3986
7.5756
5
330
600
1968.522
600
182.8784
9.0909
6
396
700
2296 . 609
700
213.3581
10.606
7
462
800
2624.695
800
243.8378
12. 121
8
528
900
2952.782
900
274.3175
13.636
9
594
1000
3280.869
1000
304.7973
I515I
10
660
2000
6561.739
2000
609.5946
30.303
20
1320
3000
9842.609
3000
914.3918
45.455
30
1980
4060
13123.48
4000
1219.189
60.606
40
2640
5000
16404.35
5000
1523.986
75 756
50
3300
6000
19685.22
6000
1828.784
90.909
60
39^
7000
22966.09
7000
2133-581
106.06
70
4620
8000
26246.95
8000
2438.378
121. 21
80
5280
9000
29527.82
9000
1
2743.175
136.36
90
5940
Digitized by
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758
SUR If EYING.
TABLE III.
Logarithms of Numbers, g 175.
i
1 0
I
»
a
6
d
Proporticmal P£irt».
.
4
5
7
S
1
1 1
1
.C170
.0^13
■°^^
1
t 3
4
6
078
0
to
,0000
.£«43
.S086
.qi^S
,oa^
0334
'O374
8 13
17
■1
15=933
11
^414
.0453
.q4(Jl
.0531
,D56q
■'^7
.. 0^.45
.utsSa
,071^
.0755
811
15
19
83 36J40.34 1
IB
.0792
.oBai
f^
.QS99
tJ93^
-09^3
-10D4
,1038
► 1071
,i(oA
7«o
M
17
»t.a4UB
3»
*3
."39
.»173
.1339
.1171
.1303
1335
■ n67
■'399
■ 1430
1 J
6
10
»3
tt
3S
«»
'^
.1461
.149*
■ 15*3
-15S3
15^4
.1614
^^644
■ <673
.1703
■»73^
1 3
6
9
la
n
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,J7«i
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.191'
■'959
.107
.30T4
6 Ri.
^4
[
t7 Bo
M
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1I
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.3D68
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t7
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■*455
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TABLES.
759
TABLE \\\.— Continued,
Logarithms of Numbers.
i
4
9
0
1
s
3
5
€
7
«
1
1
»
3
4
IE
a
T
«!7
55
74<H
74*=
-7419
.«.,
■7435
7443
-7451
■7459
■ 7466
7474
I56
748^
749^
-7497
7^5
*7S>3
.7SK
-7079
'7S3&
'7541
■7551
*l 7
57
5a
:?IU
.75^
:'^J
^7^57
■^
:JI?2
.7613
.76M
7^
7637
7701
5
59
77^
■7716
.77^3
■7731
■77S
■ 7745
775»
.77^
^7J67
7774
i
60
.778a
.7789
-7796
.7*.,
.781c
.7818
.7815
.7833
.7839
,7846
61
■7^n
.786P
.786S
75? S
.7S81
.7SSg
■S^JS
'79&3
.7910
.7917
6i
^3
79H
:l^
:iS5
7945
.B014
:i^;
-795'
El^
:^.^
.■^11
'.
64
,eSi
.8069
'S075
.8083
8089
.8096
,atoa
-8i«9
,8116
.8133
J
65
.8139
.B156
.a 143
.8149
.8i5fi
.8163
.8169
.8176
.SiB^
.81B9
6<
.815s
.«aw
.I«g
s^.s
.83*3
Swfi
8^35
8^41
J348:
8354
67
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.8i67
.8374
.8380
.83@7
.8:.^*
8^
8]to6
.Sii2
■ 83*9
63
.8335
Bj3i
.S338
-8344
.835^
•S3S7
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t
6g
-83S8
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,8431
.8437
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.84^-'
.84^6
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■8439
■8445
70
J4SI
8457
.846 J
.8470
.847*5
.848:;
.8*88
8494
Sjoq
.8506
71
.85.3
.85,19
Mn
.Bsjt
-8537
8S4?
.8549
-8555
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.8^67:
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.8657
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.866j
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.8681
.868fi
74
.869=
.8704
.S713
.8716
.87^3
.87=7
»7J3
.8739
-8745
4
75
re
.8751
87515
.876g
iiS
.J774
8779
I785
.8791
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.8814
.8830
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.8859!
I
7;
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.8S71
.8876
.888*
.8893
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,8904
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■ 89^3
178
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.89=7
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^8943
.8949
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89^5
-8971
7g
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8991
^8998
90°4
.9009
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.9030
.9^35
9c
■9<»3i
■9^j6
.90*J
-9*H7
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.9^58
90«j
9069
^9074
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.9085
^9090
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.9101
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.g^Eia
9"7
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9*33
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.9a*a
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9H^
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.9.63
93:0*;i
■9*74
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■9»99
9304
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9J5h=
■93»S
.9380
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t
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94*5
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9484
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956e
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978»;
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,9850
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9859
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9^73
.9877
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9B99
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9969
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;*Co
SURVEYING,
TABLE IIlA,
Logarithms of Sines and Tangents.
1
••
Sin.
Cos.
Tan.
Cot.
Sin. (
"os.
Tan.
Cot.
o*
0.0000
8.3419 9.
9999
8.3419
X.758X
M
z
6.4637
.0000
6.4637
3 5363
.2490
9999
.249*
•7438
59
9
.764B
.0000
.•7^2
.235a
.2561
9999
.3563
eft
3
6 9408
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6.9408
3 •059a
.3630
9999
.9631
57
4
7.0658
.0000
70658
a. 934a
.3699
9999
.3700
•7300
56
5
.1637
.0000
.1637
•8373
.3760
9999
.3767
.7333
55
6
.3412
.0000
:^l
.7581
.a832
9999
.9833
.7167
54
I
.308S
.0000
.691a
.2898
9999
.7x01
53
.3668
.0000
.3668
.6«2
.5820
.2962
9999
.9963
•7037
52
9
.4180
.0000
.4180
•3025
9999
.3oa6
•6974
5«
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.4637
.0000
.4637
•5363
.3088
9999
•3089
.69x1
50
XX
.5051
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.505«
■4949
.3»5o
9999
.3»50
.6850
.6769
%
xa
.54*9
.0000
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.457«
.3aio
9999
.3a»t
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.5777
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• 5777
.4223
.3370
9999
•3a7«
.6779
47
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.6099
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.6099
•390«
.3329
9999
•3330
.6670
46
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.6398
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•6398
.3603
.3388
9999
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4S
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.6678
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.6678
.3329
.3445
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•3446
•6554
44
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.694a
.0000 1
.6943
.3058
.350a
9999
•3503
6497
43
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.7190
.7190
.3810
.3558
9999
•3559
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42
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.74a5
.0000 1
•75*1
.a575
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9999
.3614
.6386
41
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.7648
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•a35a
9999
.3669
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40
ax
.7859
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.7860
.3140
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9999
•.^7a3
•6977
0
aa
.8061
.806a
.1938
•3775
9999
.3776
.6994
23
.8255
.0000 1
.8255
.»745
.38a8
9999
.3899
.6X7X
37
24
.8439
.0000
.8439
.1561
.3880
9999
.388X
.6119
36
as
.8617
.0000 1
.8617
.1383
•393»
9999
.393a
.6068
35
a6
.8787
.0000
.8787
.xax3
.398a
•9999
•3983
.6017
34
'2
.8951
.0000
.8951
.1049
.403a
•9999
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•5967
33
a8
.9109
.0000
.9x09
.0891
.408a
•9999
•59»7
32
39
.9261
.9a6i
.0739
.4»3»
•9999
:t;i:
.5868
3«
3o
.9408
.9409
•059'
•4«79
•9999
.5819
30
3»
•955X
.0000
•9S5»
0449
.4237
.9998
.4999
5771
a9
3»
.9689
.oooo
•9689
.0311
.4a75
.9998
.4976
.57»4
aS
33
.98aa
.0000
.9823
.0177
"♦3"
•9998
.43a3
.5677
97
34
7 995a
.0000
7.995a
a. 0048
.4368
•9998
•4370
.5630
96
35
8.0078
.0000
8.0078
i.99aa
•44M
.9998
.44»6
.5584
■5
36
.oaoo
.0000
.oaoo
.9800
.4459
•9998
.446X
.5539
»4
37
.0319
.0000
.0319
.968X
-4504
•9998
.4506
5494
a3
38
•04 15
.0000
•0435
9565
.4549
.9998
•455r
.5449
99
39
.0548
.0000
.0548
■ 945a
.4593
•9998
•4595
.4638
.5405
ai
40
.0658
.0000
.0658
•934a
•4637
.9998
.5369
ao
41
.0765
.0000
.0765
•9235
.4680
•9998
.4689
.5318
i
4a
.0S70
.0000
.0870
.9130
.47a3
•9998
•4725
•5275
43
•097a
.0000
.0972
.9028
.4765
.9998
.4767
•5233
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44
.1072
.0000
.1073
.8928
.4807
.9998
.4809
•5i9»
16
45
.1.69
.0000
.1170
.8830
.4848
•9998
.4851
.5M9
«5
46
.1265
1 .0000
.1265
.8735
.4890
•9998
.4899
.5108
»4
%
.1358
' .0000
•»359
.8641
•493"
.9998
.4933
.5067
«3
48
.1450
.0000
.1450
.8550
.407«
.9998
.4973
.5097
la
49
•»539
.0000
.IS40
.8460
.5011
.9998
.5013
• 4987
XX
50
.1627
.0000
.X637
.8373
.5050
.9998
.5053
.4947
XO
S«
•»7'3
.0000
.I7»3
.8287
.5090
.9998
.5099
■X
%
52
.»797
0.0000
.1708
.820a
.5»a9
.9998
•5«3«
53
.1880
9.9909
.i88«
.8120
.5167
•9998
.5»70
.4830
7
54
.1961
1 -9999
.1962
.8038
.5206
.9998
.5908
.479a
6
55
.2041
•9999
.3041
•7959
•5343
.9998
•5a46
•4754
5
56
.aiiQ
•9999
.2120
.7880
..saSi
.9998
•5283
.4'»7
57
58
.21.^6
• 9909
.2106
.7804
.53'8
• 9997
.53ai
•4679
.2271
.Q999
.2273
77a8
•5^55
•9997
.5358
.4649
59
2346
•Q999
.2346
1 7654
•53^J^
.0997
■5394
.4606
60
8.3419
9 9999
1 8.2419
1 • 7581
8.5428 g
■1*9^7
8.543'
I. 456?
1
Cos.
1 Sin.
1 Cot.
1 Tan.
Cos.
Sin.
Col.
Tun.
89'
samized by
^oo\
5le
TABLES.
761
TABLE \UK.^Continued,
Logarithms of Sines and Tangents.
»3
14
»5
16
»7
18
'9
n
24
25
26
27
28
29
30
3»
32
33
34
35
36
39
40
4»
42
43
44
45
46
47
48
49
5°
51
52
53
54
55
56
57
5C
59
6j
Sin. Cos. Tan. j Cot.
5428
5464
5500
5535
5605
5640
5674
5708
5742
5776
5809
5842
5875
5907
5939
5972
6003
6035
6066
6097
6128
6159
6189
6330
6250
6279
6309
633<
636I
6397
6426
6454
6483
6511
^539
6567
6595
6622
6650
6677
6704
6731
6758
6784
6810
6837
6863
6889
6914
6940
6965
6991
7016
7041
7066
7090
7"5
7140
7164
7188
9997
9997
9^7
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9997
9996
9996
9996
9996
9996
9996
9996
9996
9996
9996
9996
9996
9996
9996
99<)6
9996
9996
9996
9996
9995
9995
9995
9995
9995
9995
9995
9^>95
9995
9995
9995
9995
9995
9995
9995
9995
9994
9994
9994
9994
9994
9994
9994
9994
543»
5467
5503
5538
fIS
5643
5677
57"
5745
5779
5812
5845
5878
59"
5943
5975
6007
6038
6070
6101
6133
6163
6193
62^3
6254
6283
6313
^H3
6372
6401
6430
6459
6487
6V5
6544
6571
6599
6627
6654
6682
6709
6736
6762
6789
6815
6842
6868
6894
6920
6945
697:
6996
7021
7046
707
7096
7121
7'45
7170
7»9*
1.4569
4533
4497
4462
4427
4392
4357
4323
4289
4255
4231
4188
4»55
4122
4089
4057
4025
3993
3962
3930
3899
3868
3837
3807
3777
3746
37«7
3687
3657
3628
3599
3570
3541
3513
3485
3456
3429
3401
3373
334^
3318
3291
3264
3238
3211
3185
3158
3132
3106
:^o8o
3055
3029
3004
2979
2954
2929
2904
285.
2830
2806
Cos. Sin. Cot. I Tan
87**
3*
Sin. Cos. Tan. Cot
7188
7212
7336
7260
7283
7307
7330
7354
7377
7400
7423
7445
7468
749
7513
7535
7557
7580
7602
7623
7645
7667
7688
77x0
773 »
7752
7773
7794
7815
7836
7857
7877
7898
79x8
7939
7959
7979
7999
8019
8039
8059
8078
8098
8117
8137
8156
8175
8194
8213
8232
8251
8270
8289
8307
8326
8^5
8363
838
8400
8418
8436
9994
9994
9994
9994
9994
9994
9994
9994
9994
999
9993
9993
9993
9993
9993
9993
9993
9993
9993
9993
9993
9993
9992
9992
9992
9992
9992
9992
9992
9992
9992
9992
9992
9992
9992
9993
9991
9991
9991
9991
9991
999'
999'
9991
9991
9991
999X
9991
9990
9990
9990
9900
9990
9990
9990
9990
9090
9990
9990
9989
9989
8.7194
.7218
.7242
.7266
.7290
•73«3
•7337
.7360
■7383
.7406
.7429
•7452
.7475
•7497
.7520
•7542
.7565
•7587
.7609
•7631
.7652
.7674
.7696
•77'7
•7739
.7760
.7781
.7802
.7823
.7844
.7865
.7886
.7906
.7927
.7947
.7967
.7988
.8008
.8028
.8048
.8067
.8087
.8107
•8126
.8146
.8165
.8185
.8204
• 8223
.8242
.826
.8280
.8299
-8317
.8336
.8355
8373
.8392
.8410
.8428
8.8446
2806
2782
2758
2734
2710
2687
2663
2640
2617
2594
2571
2548
2525
2503
2480
2458
2435
24>3
2391
2369
2348
2326
9304
3383
2261
3240
3219
2x98
2177
2156
2x35
2114
2094
2073
2053
2033
2012
1^2
1972
>952
'933
'9'3
'893
X874
'854
1835
1815
1796
'777
X758
1739
1720
1701
168;;
1664
1645
1627
1608
»59<>
1572
'554
Cos. Sin. Col. j Tan
86*
Sin. I Cos. I Tan. Cot.
84369
8454
84721
8490'
8508.
8525^
85431
8560
8578
8595
8613
8630
8647
8665
8683
8699
8716
8733
8749
8766
8783
8799
8816
8833
8849
8865
8882
8898;
8914
8930
8946
8962
8978
8994
9010
9026
9342
9057
9073
9089
9104
9x19
9'35
9' 50
9166
9x3.
9196
921
9226
9241
9256
9271
9286^
930 '
93»5
9330
9345
V35Q
0374
93S8
0403! 9
99898
9989
9989
99S9
9989
9989
9989
9989
9989
9989
9989
9988
9988
9988
9988
,9988
99E8
9988
9988
9988
9988
9987
9987
9987
9987
9987
9987
9987
9987
9987
9987
9986
9986
9986
9986
9986
9986
9986
9986
9986
9986
9985
9985
9985
9985
9985
9985
9985
9985
9985
9985
9984
9984
9984
9984
9984
9984
9984
9984
9984
9983,
8446
8465
8483
8501
85x8
8536
8554
8572
8589
8607
8624
S643
8659
8676
8694
8711
8728
8762
8778
8795
88x2
8829
8845
8862
8878
8895
8911
8927
8944
8960
8976
8992
9008
9024
9040
9056
9071
9087
9103
9118
9134
9' 50
9165
9180
9196
921
9226
924
9256
9272
9287
9302
9316
933'
9346
9361
9376
93VO
9405
94
Cos. Sin. Cov Tan
554
535
5'7
499
482
464
446
428
411
376
358
34 »
324
306
289
27a
255
238
222
205
105
[089
073
1056
1040
0992
0976
0960
0944
0929
09' 3
0897
0883
0866
0850
0835
0820
0804
0789
0774
0759
0744
0728
o7'3
0698
0684
0669
0654
0639
0624
0610
0505
0580,
60'
59
58
55
54
53
52
5»
50
.1
47
46
45
44
43
42
4X
40
39
38
37
36
35
34
33
32
31
30
^t^ofegle
762
SURVEYING.
TABLE III K— Continued.
Logarithms of Sines and Tangents.
Arc.
Sin.
Df.
Ck>fc
Df.
Tan.
Df.
Cot.
Arc
Arc.
Sin.
Df.
Con.
Df.
Tan.
Df.
Cot
Arc
e /
5 0
xo
ae
8.9403
X43
»37
«?4
9.9983
.998a
.9981
8.94ao
•9563
.970X
M3
138
135
T.0580
•0437
.0399
0 /
850
50
40
0 f
'5 0
xo
90
94»3o
•4177
•4823
i
•9843
9.4a8x
SO
SO
49
o.57>9
.S669
.56x9
e t
750
50
40
30
50
.98x6
8.9945
9.0070
za9
"5
zaa
•9980
•9979
•9977
9.0093
130
X37
"3
.0164
X.0034
0.9907
30
20
xo
30
40
50
.4269
•43»4
•4359
45
45
44
•983a
.4430
.4479
•4537
H
SSTO
.5521
.S473
30
ao
to
60
10
ao
.oz9a
.0311
.04a6
X19
"5
"3
.9976
•9975
•9973
a
.oai6
.0336
•0453
I30
"7
"4
.9664
•9547
840
50
40
16 0
xo
30
•4403
•4447
.449«
44
44
4a
.9838
.9835
.9831
.4669
47
47
47
•533*
740
SO
40
30
50
.0755
X09
Z07
104
.997a
•997»
.9969
B
itx
X08
105
•9433
.9322
.9314
30
ao
xo
30
40
50
.4533
43
4a
4»
.9817
.98.4
.9810
.47x6
.4763
.4808
45
.519a
30
ao
xo
7 0
zo
ao
.0961
.1060
zoa
99
97
.9968
.9966
•9964
.0891
•0995
.X096
X04
lOX
98
.9x09
.9005
.8904
830
50
40
X7 0
xo
ao
.4659
.4700
•4741
4»
.9803
.9798
•4943
45
45
44
•5147
.510a
.5057
730
SO
40
30
40
SO
"57
.xasa
.»345
95
93
9>
.9963
.9961
•9959
.XX94
97
94
93
.8806
.8709
.86x5
30
30
zo
30
40
50
•4781
.483Z
.486X
40
40
39
.9794
.4987
.5031
.5075
44
44
43
.4969
.49aS
30
90
xo
80
10
ao
■1436
.xsas
.z6xa
1
•9958
•9956
•9954
..478
87
.8533
.8431
.834a
83 0
50
40
x8o
xo
so
•4900
•4939
•4977
1
.97^
.9778
.9774
.5»8
.5x6x
•5*03
43
4a
4a
.488a
.4839
• 4797
Tao
SO
40
30
40
50
:5^
.1863
s
80
.9952
•9950
•9948
•«9«5
86
.8085
10
30
40
50
.So«5
• SOSa
.5090
i
.9770
•976s
.9761
•53*9
4«
4a
4»
.4755
30
ao
zo
90
xo
ao
•'9 ♦3
.ao^2
.aioo
I
•9946 a
.»44' a
•994a a
.1997
.ao. J
.3158
81
80
78
.8003
.7933
.784a
81 0
50
40
X9 0
10
so
.5xa6
• Si'^3
.5199
i
•9757
•97Sa
.9748
•S370
•54"
•S45«
4»
40
40
•4549
7X 0
SO
40
30
40
50
.ax76
.aasi
3324
75
73
73
.9940 a
.99 J8 2
.9^36 a
.3336
.3313
.3389
5?
74
•7687
.7611
30
so
10
30
40
50
•5235
.5370
.5306
35
9743
•9739
.9734
•549»
•553«
•S57»
40
40
40
.4509
.4469
•4429
30
ao
zo
10 0
zo
ao
•2538
71
70
68
9931 3
.91)31 a
.9929 2
.3463
•2536
.2609
73
73
71
.7537
.7464
•739<
80 0
50
40
20 0
xo
so
•5341
•5375
•5409
34
34
34
.9730
-97a5
.9731
.561X
39
.4389
•4350
•43"
700
50
40
30
45
50
.a6o6
.3674
.2740
68
66
66
•9927, 3
•9924 a
•99a3 3
.2680
.2750
.38x9
i
.7330
.7250
.718X
30
20
xo
30
40
so
•5443
-5477
•SS'o
34
33
33
.97x6
.5737
i
.4»;3
.4»34
.41^6
30
ao
10
II 0
10
20
.3806
.2870
•2934
63
.9919
.9917
•99M
a
3
.3887
.a953
.3020
66
67
65
.7i»3
79 0
50
40
ai 0
10
30
.5543
.5576
.5609
33
33
3a
■E
.969a
-584a
•5879
.59«7
1?
37
•4158
.4121
•4083
€4f 0
so
40
.10
40
5^
••3"9
61
•99»a, 3
•9909, a
.9907 3
•3085
•3H9
.3212
64
63
M
30
20
10
30
40
50
.5641
• 5673
•5704
3a
3»
3a
•9677
.5954
37
.4046
•4009
•3972
30
ao
zo
ta 0
lO
20
•-1
.3296
57
•9904 3
.9901 a
.9899I 3
.3275
•3336
•3397
61
61
61
.6725
.6664
.6603
780
50
40
33 0
zo
SO
•57<^
.5798
3»
31
30
.9661
.6064
.6100
.6x36
i
•393«
680
50
40
30
40
50
•3353
•3410
.3466
55
.9896, 3
•9893 3
.9890 3
.3458
.3517
•3576
59
It
•6424
30
30
10
30
40
50
.5828
•1^
3»
30
30
.9^56
.9646
.6173
.6908
.6*43
36
.3838
•379a
•3757
30
ao
10
•3 0
10
20
.3521
.3629
54
54
53
•95?7 3
•9??» 3
.9831 3
.3691
.3748
57
11
.6366
.6309
.6353
77 0
50
40
23 0
10
30
•5978
39
30
39
.9640
9635
.9629
.6379
35
34
35
•3«52
670
50
40
30
.36S2
5*
5a
51
.9878' 3
.9875 3
.9872 3
.3804
.3859
.39M
11
54
.6196
.6141
.6086
30
30
zo
30
40
SO
.6036
.6065
39
.96a4
.9618
•96^3
6
1
34
3S
34
.36.7
30
ao
10
14 0
10
ao
••^sJJ
•3937
50
50
49
.9863 4
-3968
.4031
•4074
53
53
53
.6033
•5970
.5936
76 0
53
24 0
xo
30
.6093
.6X31
.6x49
38
38
38
.9607
.9603
.9596
I
6
34
33
34
•3447
€60
50
40
3'
40
50
• 3)36 49
4035I 48
•4»8i 47
•9359 3
.9856 3
•9853 4
.4127
.4x78
.4230
52
5*
5«
•5873
.5823
•5770
30
20
10
30
40
50
.6177
.6305
.0333
•28
37
37
•9590
•9584
•9579
6
1
.«54
33
34
33
.34J3
•3346
30
ao
■0
»5 0
9-4«3o| 47
9-9349, 3
9-4281
50
0.571975 0
'5 0
0.6239
27
dT
9-9573
7
9.6687
J3
o-33«3
650
Arc
Arc
Coa. 'Dt
Sin. Dt
Cot. .Df.
Tan. lArc.
Arc.
Cm.
Sin.
DfT
Cot.
DC
Tan.
TABLES.
7^-6
TABLE lUK'-QmHnued.
Logarithms of Sines and Tangents.
▲ra.
Sin.
M.
Ooa.
Df.
Tan.
Df.
Ook.
Are.
Arc
Bin.
Df.
Oofc
Df.
Tml
Df.
Oot
Arc.
to
30
.6313
37
27
97
.956X
6
6
6
9.6687
.6730
.675a
33
3a
33
0.3313
.3280
• 3248
e f
650
SO
40
0 /
35 0
xo
ao
18
x8
x8
>-9»34
.9x35
.9x16
9
9
9
^.8452
27
27
27
0.1548
.1521
•«494
55 0
SO
40
30
50
1^
.639a
76
36
a6
•9SSS
.9549
•9543
6
6
6
.6785
.6817
.6850
3a
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SURVEYING.
TABLE v.*
Horizontal Distances and Elevations from Stadia Readings. § 204.
r
Mlnntes.
00
1
L<>
20
»
Hor.
Diff.
Hor.
DiflC
Hor.
DiflT.
Hor.
Diff.
Disl.
Elev.
Dist.
Elev.
Dist.
Elev.
Dist.
Elev
0 . .
100.00
aoo
99-97
1.74
99^
3-49
99-73
5-23
2 . .
**
0.06
**
1.80
99-87
3-55
99-72
5.28
4 . .
««
0.12
«(
1.86
(•
3.60
99.71
5-34
6 . .
«•
0.17
99.96
1.92
•'
3-66
44
540
8 . .
M
0.23
M
1.98
99^
3-72
99.70
5-^6
10 . .
««
0.29
14
2.04
w
zn^
99.69
5.52
12 . .
M
0.35
«t
2.09
99-85
3-84
«<
5-57
14 . .
M
041
99-95
2.15
41
3.90
99.68
5.63
16 . .
M
0.47
i«
2.21
99-84
3-95
44
5-69
18 . .
M
0.52
«
2.27
«4
4X>i
99-67
5.75
20 . .
«
asS
M
2-33
99-83
4.07
99.66
5.80
22 . .
<4
a64
99-94
2.38
44
4.13
44
5.86
24 . .
«
0.70
M
2.44
99-82
4.18
99.65
5.92
26 . .
99-99
0.76
«
2.50
44
4.24
99.64
5.98
28 . .
<t
* 0.81
99-93
2.56
99.81
4.30
99-63
6.04
30 • •
M
0.87
««
2.62
44
4-36
a
6.09
32 . •
«<
0.93
44
2.67
99.80
4.42
99-62
6.15
34 . .
M
0.99
<4
2.73
M
4.48
w
6.21
36 . .
M
1.05
99.92
2.79
99-79
4-53
99.61
6.27
38 . .
<t
I. It
44
2^5
14
4.59
99.60
6.33
40 . .
M
1.16
M
2.91
99-78
4.65
99-59
6.38 ;
42 . .
M
1.22
99.91
2.97
44
4.71
44
644
44 . .
99.98
1.28
44
3.02
99-77
4.76
99.58
6.50
46 . .
**
1.34
99.90
3-08
44
4^2
99-57
6.56
48 . .
«f
140
44
3-14
99.76
4.88
99.56
6.61
50 . .
U
M5
41
3.20
44
4.94
44
6.67
52 . .
4<
1.51
99-89
3.26
99-75
4.99
99.55
6.73
54 . .
*i
'•57
44
3-3»
99-74
5-05
99-54
6.78
56 . .
99-97
1.63
44
3-37
44
5.11
99-53
6.84
58 . .
**
1.69
99-88
•3-43
99.73
5-»7
99-52
6.90
60 . .
^-0.75
r= i.oo
ir=1.25
1.74
44
3-49
44
0.75
1.00
5-23
0.03
ao4
99-5'
6.96
0.75
1.00
0.01
0--5
0.02
0-75
0.05
0.0 1
1.00
0.03
1.00
0.06
125
0.02
I.2S
0.03
1.25
0.05
1.25
0.08
♦ This uble was computed by Mr. Arthur Winslow of the Sutc Geological Sarver of
sylTftDia. See also Colby's Slide Rule, p. a6s.
TABLES.
773
TABLE V,— Continued,
HouzoNTAL Distances and Elevations from Stadia Readings.
40
5«
60
70
Minates.
Hor.
Diff.
Hor.
Diff.
Hor.
Diff.
Hor.
Diff
Dist.
Ekv.
Dist.
Elev.
Disi.
Elev.
Dist.
Elev.
O . .
99.5^
6.96
99.24
8.68
98.91
10.40
98.51
12.10
2
<i
7^)2
99.23
8.74
98.90
10.45
98.50
12.15
4
99-50
7.07
99.22
8.80
98.88
10.51
98.48
12.21
6
99-49
7'^Z
99.21
8.85
98.87
'O.57
98.47
12 26
8
99-48
7.19
99.20
8.91
98.86
10.62
98.46
12.32
ID
99.47
7.25
99.»9
8.97
98.85
10.68
98.44
12.38
12
9946
7.30
99.18
9-03
98.83
10.74
98.43
1243
14
4i
7.36
99.17
9.08
98.82
10.79
98.41
12.49
i6
99.45
7.42
99.16
9.14
98.81
10.85
98.40
12.55
i8
99.44
7.48
99-15
9.20
98.80
10.91
98.39
12.60
20
99-43
7.53
99.14
9.25
98.yS
ia96
98.37
12.66
22
99-42
7-59
99.13
9-3«
98.77
11.02
98.36
12.72
24
9941
7.65
99.11
9.37
98.76
11.08
98.34
12.77
26
99.40
7-7^
99.10
9.43
98.74
11.13
98.33
12.83
28
99.39
7.76
99.09
9.48
98.73
II. 19
98.31
12.88
30
9938
7.82
99-08
9.54
98.72
11.25
98.29
12.94
32
99.38
7^
99.07
9.60
98.71
11.30
98.28
13.00
34
99-37
794
99.06
9.65
98.69
11.36
98.27
'3.05
36
9936
7-99
99.05
9.71
98.68
11.42
98.25
i^u
38
9935
805
9904
9.77
98.67
11.47
98.24
I3.»7
40
99-34
8.11
99-03
9.83
98.65
"53
98.22
13.22
42
99.33
8.17
99.01
9.88
98.64
11.59
98.20
13.28 1
44
99.32
8.22
9900
9.94
98.63
11.64
9819
^y^i
46 .
99-31
8.28
98.99
10.00
98.61
11.70
98.17
U.39
48
99.30
8.34
98.98
10.05
98.60
11.76
98.16
13.45
SO .
99.29
8.40
98.97
10. II
98.58
11.81
98.14
^3.50
52 .
99.28
8.45
98.96
10.17
98.57
11.87
98.13
13.56
54 ■
99.27
8.51
98.94
10.22
98.56
n.93
98.11
13.61
56 .
99.26
8.57
98.93
10.28
98.54
11.98
98.10
13.67
58 .
99-25
8.63
98.92
10.34
98.53
12.04
98.08
13.7.^
60 . .
r=zo.75
^= IXX)
99.24
8.68
98.91
10.40
98.51
12.10
0.08
98.06
0.74
13.78
0-75
0.06
0.75
0.07
0.75
0.10
IJOO
o.cS
0.99
0.09
0.99
0.1 1
0.99
0.13
>
^ = 1
•25
1.25
0.10
1.24
CM
1 24
o.i4)i
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74
SURVEYING.
TABLE V , --Continued.
Horizontal Disiances and Elevations from Stadia Readings.
8
;o
90
100
110 1
Bflnotes.
"
Hor.
Diff.
Hor.
DUt
Hor.
Difr.
Hor.
DHL
Disu
Elev.
DUt.
Ekv.
Dist.
EkY.
Dist.
Eler.
o . .
98.06
13.78
97.55
>'S45
96.98
17.10
96.36
18.73
2
98.05
13.84
97.53
15.51
96.96
17.16
96.34
18.78
4
98.03
'>89
97.52
15.56
96.94
17.21
96.32
18.84
6
98.01
"3.95
97.50
15.62
96.92
17.26
96.29
18.89
8
98.00
14.01
97.48
15.67
96.90
^ 17.32
96.27
18.95
to
97.98
14.06
97.46
15.73
96.88
17.37
96.25
19.00
12 .
97.97
14.12
9744
15.78
96.86
1743
96.23
19.05
M
97.95
14.17
9743
1^84
96A|
1748
96.21
19.II
i6
97.93
14.23
97.41
15.89
96.82
17.54
96.18
19.16
i8
97.92
14.28
97.39
"5.95
96A)
17.59
96.16
19.21
20
97.90
14.34
97.37
16x0
96.78
17.65
96.14
19.27
22
97.88
14.40
97.35
16.06
96.76
17.70
96.12
19-32
24
97.87
14.45
97.33
16.11
96.74
17.76
96.09
19.38
26
97.85
.14.51
97.31
16.17
96.72
17.81
96.07
1943
28
97.83
14.56
97.29
16.22
96.70
17.86
96.05
19.48
30
97.82
14.62
97.28
16.28
96.68
17.92
96.03
19-54
32
97.80
14.67
97.26
16.33
96.66
.17.97
96.00
19-59
34
97.78
14.73
97.24
16.39
96.64
18.03
95.98
19.64
36
97.76
14.79
97.22
16.44
96.62
18.08
95.96
19.70
38
97.75
14.84
97.20
id50
96.60
18.14
95.93
19.75
40
97.73
14.90
97.18
16.55
96.57
18.19
959"
19.80
42
97.71
14.95
97.16
16.61
96.55
18.24
95.89
19.86
44
97.69
15.01
97.14
16.66
96.53
18.30
95.86
19.91
46
97.68
15.06
97.12
16.72
96.51
18.35
95.84
19-96
48
97.66
15.12
97.10
16.77
96.49
1841
95.82
20.02
50
97.64
15.17
97.08
16.83
9647
1846
95-79
20.07
52
97.62
i5-23
97.06
16.88
96.45
18.51
95-77
2ai2
54
97.61
15.28
97.04
16.94
96.42
18.57
95.75
2a 18
56
97.59
15-34
97.02
16.99
9640
18.62
95.72
2a23
58
97.57
15.40
97.00
17.05
96.38
18.68
95.70
2aa8
60 . .
97.55
1545
96.98
17.10
0.12
96.36
18.73
95.68
20.34
ir = 0.75
0.74
0.11
0.74
0.74
ai4
0.73
0.15
ez=. 1.00
0.99
0.15
0.99
ai6
a98
ai8
a98
0.30
r= I
•25
J.23
/
ai8
1.23
a2i
1.23
a23
1.22
ais 1
TABLMS.
rrs
TABLE V.'-CmHnmid.
BoRBBOin'iiL Distances iUCD Elevations from Stadia Rbadinos.
120
13«
14*
15«
IObvm.
Kor.
Dur.
Hor.
Dim
Hor.
Diff.
Hor.
Diff.
Dist
EleT.
DUt
Ekv.
Dist.
Ekv.
Dut.
EleT.
0 . .
95.68
2a34
94.94
21.92
94.15
23.47
93-30
25.00
a
95-65
20.39
94.91
21.97
94.12
23-52
93-27
25.05
4
9563
20.44
94.89
22.02
94-09
23.58
93-24
25.10
6
95.61
20.50
94.86
22.08
94.07
23.63
93.21
25.15
8
95-58
20.55
94.84
22.13
94.04
23.68
93.18
25.20
10
9556
2a6o
94.81
22.18
94.01
23.73
93.16
25.25
ts
95-53
2066
94.79
22.23
9398
23.78
93.13
25.30
14
95-51
20.71
94.76
22.28
93-95
23-83
93.10
25.35
16
9549
20.76
94.73
22.34
93-93
23.88
9307
2540
i8
95.46
20.81
94.71
22.39
93-90
23.93
93-04
2545
so
95-44
20.87
94-68
22.44
93.87
23.99
93-01
25.50
ss
9541
20.92
94.66
22.49
93-84
24.04
92.98
25-55
S4
95-39
20.97
94.63
22.54
93-8i
24.09
92.95
25.60
s6
95-36
21.03
94.60
22.60
93-79
24.14
92.92
25.65
s8
95-34
21.08
94.58
22.65
93-76
24.19
92.89
25.70
y>
95-3*
21.13
94.55
22.70
93-73
24.24
92.86
25.75
3«
95-«9
SI.18
94.52
2^.75
93-70
24.29
92.83
25.80
34
95-«7
21.24
94.50
22.80
93.67
24.34
92.80
25.85
36
95-24
21.29
9447
22.85
93.65
24-39
92.77
25.90
3»
95.22
21.34
9444
22.91
93.62
24.44
92.74
25-95
40 .
95-19
21-39
9442
22.96
93-59
24.49
92.71
2600
4S .
95-17
21.45
94.39
23.01
93.56
24.55
92.68
2605
44
95-M
21.50
94.36
23.06
93.53
24.60
92.65
26.10
46
95.12
21.55
94-34
23.11
93-50
24.65
92.62
26.15
48
95-09
21.60
94.31
23.16
9347
24.70
92.59
26.20
SO .
95-07
21^
94.28
23.22
9345
24.75
92.56
26.25
P
95-04
21.71
94.26
23.27
9342
24.80
92.53
2630
54
95.02
21.76
94-23
2332
93-39
24.85
92.49
26.35
S6
94.99
21.81
94.20
23-37
93-36
24.90
92.46
2640
58
94.97
21.87
94.17
2342
93-33
24.95
9243
26.45
60 . .
*»<X75
#S8l^0O
94-94
21.92
94.15
2347
9330
25.00
9240
26.50
0.73
ai6
0.73
0.17
0.73
0.19
0.7s
0.20
098
a22
0.97
0.23
0.97
0.25
0.96
a27
«si
•»5
I.2S
0-27
1.21
a29
1.21
0.31
1.20
••34
^7^
SURVEYING.
TABLE \ .--Contitttiid.
Horizontal Distances arix ^lbvations from STAbiXvKsxoam.
I
I60
170 '
iso -^-J
,». j
Minnies.
■
Hor.
DtflT.
Hor.
Diff.
Mdr.
XM.
.'- Hor,
Diff. ;
Dist.
Eldv.
Dttt.
Elev.
Di»t.
•••EWv.
' Dist
Ekv. I
0 . .
92.40
26.50
9M5
27.96
90-15
29-39
-89-40
30.78
2 . .
9237
2d55
91.42
28.01
9042
29.44
89.36-
:!^^i
4 . .
9234
26.59
91.39
28.06
90.38
29.48
89.33
3^7
6 . .
92-31
26.64
9'.35
28.10
90.35
29.53
^.29.
3^.92
S . .
92.28
26.69
91.32
28.15
90.31
29^58
^ ^.26.
3^.97
16 . .
92.25
26.74
91.29
28.20
90.28
29.62
: %22
• 3t*>i
12 . .
92.22
26.79
91.26
28.25
90.24
29.67
5^.18-
. 3»to6
:i4 . .
92.19
26.84
91.22
28.30
90.21
- 29.72
1^15-
• 3»lio
i6 . .
92.15
26.89
91.T9
28.34
90.18
2^76
€^.ii-
•3*^5
i8 . .
92.12
26.94
91. t6
28.39
90.14
2^81
^.08'
■ 3*.»9
• 26 . .
92.09
26.99
91.12
28.44
90.11
29;86
«9.04
3»*4
32 . .
92.06
27.04
91.09
28.49
90.07
29.90
". fl^oo
3tfe8
24 . .
92.03
27.09
91.06
28.54
90.04
29.95
^ 88.96-
•3«4J
26 . .
92.00
37.13
91.02
28.58
90.00
30.00
88.^3
3**^
28 . .
9«-97
27.18
9<>.99
28.63
89.97
30^04
'«8.89
3*5^2
30 . .
9' -93
27.23
90.96
28.68
89.93
3«J.o9
88.86^
3»«47
32 • •
91.90
27.28
90.92
2873
89.90
3^*14
- S8.82
• 3<^i
34 . .
91.87
27.33
90.89
28.77
89.86
*>.i9
• «8.78
3«^6
36 . .
91.84
27.38
90.86.
28.82
89.83
pn
«875
3*a6o
38 . .
91.81
27.43
90.82
28.87
89.79
• J0.2S
«7i
'3*^5
40 . .
9' 77
27.48
9079
28.92
89.76
3<*.32
'- «.67'
• 3«*9
42 - .
9«74
27.52
90.76
28.96
89^72
3A37
^.64
3**74
44 . .
9' 71
27.57
90.72
29.01
89.69
30-4»
S8.60
3*^8
46 . .
91.68
27.62
90.69
29U06
89.65
30.46
^.56
J<«3
48 . .
91.65
27.67
90.66
29.11
89.61
30.51
88.53-
3^«7
50 . .
. 91.61
2772
90.62
29.15
89.58
30.55
88.49'
3*^2'
52 . .
9t.58
2777
90-59
29.20
89.54
jd.6o
«8.45
3«96
54 . .
9t.5S
27.81
90-55
29,25
89.51
jO.65
88.41
3*i>i
56 . .
^.52
27.86
90.52
29.30
89.47
'30.69
88.38
3^5
58 . .
91.48
27.91
90.48
29.34
89.44
3*74
88.34
3«09
60 . .
^ = 075
rt= i.oo
r*=i.25
9>45
27.96
90.45
2939
89.40
jo.78
88.30
3*>.4
072
0.21
0.72
0.23
071
0.95
0.24
0.32
0.71'
■
0.96
0.28
0.95
0.30
-0.94-
1
1.20
0.35
1.19
0.38
I.19
. 0.40
5 : i.iSc
TABLES.
777
TABLE \ .—Continued.
Horizontal Distanci
:S AND ]
2:
Elkvaiions from Stadia Readings.
Mlnntet.
200
10
r
2P
280
Hor.
D«r.
Hor.
DiflC
Hot.
Diff.
Hor.
Dift.
Dbu
Ekv.
Dist.
filer.
Dist
filer.
Dist.
EI.V.
O . •
88.30
32.»4
87.16
33.46
85.97
34-73
84.73
35-97
2 . ,
88.26
32.18
87.12
33.50
85.93
34.77
84.69
36.01
4 . .
88.23
32.23
87.08
33-54
85.89
34.82
84.65
36.05
6 . .
•88.19
32.27
87.04
33-59
85.85
34.86
84.61
36.09
8 . .
88.15
32.32
87.00
33.63
85.80
34.90
84-57
36.13
10 . .
88.11
32.36
86.96
33.67
85.76
34.94
84.52
36.17
12 . .
88.08
32.41
86.92
iz-n
85.72
34.98
84.48
36.21
14 . .
88.04
32.45
86.88
33-76
85.68
35.02
84.44
36.25
i6 . .
88.00
3249
86.84
33.80
85.64
35.07
84.40
36.29
i8 . .
87.96
32.54
86.80
33.84
85.60
35.11
84.35
3633
20
87.93
32.58
86.77
3389
85.56
35.15
84.31
36.37
22 . .
87.89
32.63
86.73
33.93
85.52
35-»9
84.27
36.41
24 . .
87.8s
3267
86.69
33.97
8548
35.23
84.23
36.45
26 . .
87.81
3272
86.65
34.01
85.44
35.27
84.18
3649
28 . .
87.77
32.76
86.61
34.06
8540
353'
84.14
36.53
30 • .
87.74
32. So
86.57
34.10
85.36
35.36
84.10
36.57
32 • •
87.70
32.85
86.53
34.M
85.31
35-40
84.06
36.61
34 . .
87.66
32.89
86.49
34.18
85.27
35.44
84.01
36.65
36 . .
87.62
32.93
8645
34.23
85-23
35.48
8397
36.69
38 . .
87.5S
32.98
86.41
34.27
85.19
35.52
83.93
36.73
40 . .
87.54
33.02
86.37
34.31
85.15
35.56
83.89
36.77
42 . .
87.51
33.07
86.33
34.35
85.11
35.60
83.84
36.80
44 . .
87.47
33."
86.29
34.40
85.07
35-64
83.80
36.84
46 . .
87.43
33- » 5
86.25
34-44
85.02
35.68
83.76
36.88
48 . .
87.39
33.20
86.21
34.48
S4.98
3572
8372
36.92
50 . .
87.35
33.24
86.17
34.52
84.94
35.76
83.67
36.96
52 . .
87.31
33.28
86.13
34.57
84.90
35.80
83.63
37-00
54 . .
87.27
33.33
86.09
34.61
84.86
35.85
83.59
37.04
56 . .
87.24
33-37
86.05
34.65
84.82
35.89
83.54
37.08
58 . .
87.20
33.41
86.01
34.69
84.77
35.93
83.50
37.12
60 . '.
^'=075
r = 1.00
r=i.25
87.16
33-46
85.97
0.70
34-73
0.27
0.37
84.73
35-97
83.46
37.16
0.70
0.26
0.69
0.29
0.69
0.30
0.94
0.35
0.93
0.92
0.38
a92
0.40
1.17
0.44
1.16
0.46
1.15
0.48
1.15
0.50*
.
77^
yjRV EYING.
TABLE V.-^CotUinued,
Horizontal Distancss and Elevations prom Stadia Rkadings.
gjt)
in
95-0
31-1
tSo
€11
fSo
til
00*1=^
SZ-o = j
• • 09
iro
68-0
St'X)
6g*o
rto
zr-o
06*0
ito
160
SC-o
99-0
Z90
890
i€o
89-0
Stit'
96ZZ
Sfrot-
6e-6Z
ot-ee
8^*08
o€ge
tiTg
et'-ifr
lo-gZ
efrofr
tt-6Z
9C-6e
^808
92-8^
8»28
• • 85
ee-it
90*8^
gC-ot
8>-6Z
ee-6c
^8*08
erge
Zz'z%
• • 95
se-i>
ov%L
SCof
fS-6Z
6z-6e
r6og
6ige
Iz'zS
• • ts
te-it'
Srg^
i€o^
85-6Z
92-6C
Z6og
sige
re?g
• • tS
6rit
or-giC
gzofr
J9-<5v.'
2j-6e
10- 1 g
irge
9C-rg
• • oS
Qrit'
SrgZ
tzofr
Z9-6Z
8i-6e
90-ig
80-8t
itrg
• - 8^
rzifr
oe-8^
izofr
rZ-6Z
S16C
orig
to-8e
Strg
• • 9t
en*-
^•gZ
grofr
9Z6Z
ir6e
Srig
oo-ge
6t-rg
• • tt
9rit
ee-gz
trot
18-6^
80-6C
6iig
96Z€
tSTg
• • rt
rri*-
H-g^
not
98^Z
to-6e
tr-ig
e6Z£
85-^8
• • ot
eoi*-
6>'giC
Zo-ot
C6-6Z
00-6C
82-18
6g-ze
e9Tg
• • 8C
90- It'
i^-g^
toot
S6-6Z
Z6-8e
ee-ig
sg-ze
Z9«g
• • 9C
ro'ifr
8S-8^
oo-ot
OO-og
e6-8e
8f-«8
ig-ze
zl'Z2
• • t€
66-0^
e9-gz
Z6-6e
toog
68-8e
«t-Ig
U'LZ
9l'zs
• • t£
96ofr
89*8^
e6-6C
6oog
98*8^
Zt-ig
H'l£
og-28
• • oC
z6ot
€^•8^
c6'6e
trog
29*8^
iSig
oz-ze
58-28
• • 8*
68-ofr
^/•8^
98-6C
81-08
8^-8^
9S-ig
99ze
68-28
• • 9S
98'0«'
^8*8^
e8^e
Cr-og
sz-ge
09-18
r9-ZC
e6rg
• • ¥z
ryo^
^8-8^
6z-6e
82*og
iz-ge
S9ig
gs-ze
86-28
• • zz
6Z-ofr
26-gZ
9Z-6f
2C-og
^•8C
69ig
ts-ze
roCg
• • oc
9Z-o^
96-gZ
ZZ-6C
ze-og
>9-8e
tZ-ig
iS-ZC
zo-eg
• • 81
c^o*-
I0-6Z
69-6e
it-og
09-8^
8^-18
if'it
irCg
• • 91
690!'
90-6Z
S9-6C
9t-og
9^-8e
^8*18
£\rl£
Si-€g
• • H
990*'
II-6Z
I9-6C
I Jog
esge
^8-^8
6eze
oj-Cg
• • CI
C9'0fr
SI-6Z
85-6e
SSog
6t-ge
t6*ig
seze
tz-Cg
• • 01
es-ot'
0C-6Z
tS-6C
090g
st-ge
96ig
leze
82-eg
' • 8
SS-ot
Sj-6i:
iS-6C
S9og
it-ge
lo-rg
IZ'lt
eecg
• • 9
rSot' ,
oe-6Z
Zt-6C
690g
8e-8€
S02g
ivLZ
ZC-€g
• • ¥
6frofr
h:-6z
tt-6C
tZ-og
t€-ge
602g
oi'Li
iKg
• • z
^)rd^
6C-6Z
ot-6e
8^o8
oe-ge
trrg
9rZe
9*^8
• • 0
•A»ia
•ISIQ
•A»ia
•ISIQ
A»ia
i«a
•Aaia
»K1
•JBKI
•40H
'JtfKI
•JOH
•jtfja
•40H
jwa
•40H
•— MHifTt
oI
z
oOS
o'QZ
Digitizao^W^jOC
gle •
TABLES.
779
TABLE V , ^Continued,
Horizontal Distancss and Elevations from Stadia Readings.
«8o
290
aoo
Mtamtos.
Hot.
DiflT.
Hor.
Diff.
Hor.
Diff.
Disc
Elev.
Dist
ElCY.
DUt.
Elev.
O . .
77.96
41.45
76.50
42.40
75.00
43-30
2
7791
41.48
7645
4243
74.95
43-33
4
77.86
41.52
7640
42.46
74.90
43.36
6
77.81
41.55
76.35
42.49
74.85
43-39
8 .
77.77
41.58
76.30
42.53
74.80
4342
10
77.72
41.61
76.25
42.56
74-75
4345
12
77.67
41.65
76.20
42.59
74.70
4347
14
77.62
41.68
76.15
42.62
74.65
43-50
i6
77.57
41.71
76.10
42.65
74.60
43-33
i8
77.52
41.74
76.05
42.68
74.55
43-56
20
77.48
41.77
76.00
42.71
7449
43.59
22
77.42
41.81
75-95
42.74
74.44
43.62
24
77.38
41.84
75-90
42.77
74.39
43.65
26
77.33
41.87
75-85
42.80
74.34
43-67
28
77.28
41.90
75.80
42.83
74.29
43.70
30
77.23
41.93
75.75
42.86
74.24
43.73
32
77.18
41.97
75-70
42.89
74.19
43.76
34
77.13
42.00
75-65
42.92
74.14
43-79
36
77.09
42.03
75-60
42.95
74.09
43-82
38
77.04
42.06
75-55
42.98
74.04
43-84
40
76.99
42X)9
75-50
43.01
73-99
43.87
42
76.94
42.12
75-45
43-04
73.93
43.90
44
76.89
42.15
75-40
43-07
73-88
43-93
46
76.84
42.19
75-35
4310
73.83
43-95
4S
76.79
42.22
75.30
4313
73-78
43.98
50
76.74
42.25
75.25
43.16
73-73
44.01
52
76.69
42.28
75.20
43.18
73.68
44.04
54
76.64
42.31
75-15
43-21
73.63
44.07
56
76.59
42.34
75.10
43-24
73.58
44.09
5S
76.55
42.37
75-05
43.27
73.52
44.12
60 . .
I = 0//S
r=IiX>
76.50
42.40
a36
75.00
43-30
73-47
44.15
0.66
0.65
0.37
0.65
0.38
088
0^8
0.87
0.49
0.86
D.51
rsi
•as
I.IO
a6o
i.a>
0.62
D^jijj^db
78o
SURVEYING.
TABLE VI.
Natural Sines and Cosines.
/
"o
0
V I
20 ,
30 II 40
Sine
.01745
Cosin
.99965
Sine
.03490
Cosin
Sine
Cosin' Sine |Cosin
1
.99939
.06234
.99863!
.06976
.99756
60
1
.u
.01774
QOQftii
.03519
.99938
.05263
.99861
.07005
.997541 50
2
.0
.01803
.99964
.03548
.99937
.06292
.99860'
.07034
.997521 58
8
.0
.01832
.99963
.03577
.99936
.05321
.998581
.07063
.99750,57
4
.0
.01862
.99983
.08606
.99935
.06350
.99857
.07092
.99748, 56
6
.0 ;;
.01891
.99982
.03635
.99934
.06379
.998651
.on2i
.99746 55
6
.o-.^sv-j
UlU.t.
.01920
.99962
.08664
.99933
.05408
.99654
.07150
.99744
54
7
.^y^^
tlllt^.
.01949
.99981
.03693
.99982
.05437
.998521
.07179
.99742
58
8
.^ym,
(JllO.
.01978
.99980
.03?^
.99931
.05466
.99851
.0ra08
.99740
58
0
.OOi^
Ori«*
.02007
.99980
.03752
.99930
.99929
.06495
.99649
.07287
.99788
51
10
.OCSSOl
Oris.
.02036
.99979
.08781
.05524
.99817
.07266
.99786
50
11
.On*5»
,wyn
.02065
.99979
.03810
.99927
.06663
.99646
.07295
.99784
49
12
.0.':^]^^
.iJ'f:f''9
.02094
.99978
.03839
.99926
.06682
99644
.07324
.99'<-31 48
13
.Oin:s
/.M':»'j9
.02123
.99977
.03868
.99925
.06611
!99642
.07358
.99729; 47
14
.0.^1117
.'.h>:J''9
.02162
.99977
.03897
.99924
.06640
.99841
.07382
.99727' 46
15
.0«m:;i;
.!H.|'J'r9'
.02181
.99976
.03926
.99923
.06669
.99689
.07411
.99725' 45
16
.0<iiij:i
,^f:KlJii9'
.02211
.99976
.03955
.99922
.05698
.99638
.07440
.99723' 44
17
.0«'iirj
.■.nn^'i9'
.02240
.99975
.03984
.99921
.05727
.99686
.07469
.99721148
18
.Oii.v.'j
li^^li'l^'
.02269
.99974
.04013
.99919
.06756
.99634
.07496
.99719142
19
.^\:uc\
.IW.13:
.02298
.99974
.04042
.999181
.05785
.99683
.07527
.99716, 41
80
.0o5b;i
.ijyL<i8i
.02327
.99978
.04071
.99917
.05814
.99681
.07556
.99714 40
21
.00611
,9CaiS
.02366
.99972
.04100
.99916
.05S44
.99629
.07586
.99712189
22
.frKViO
.>p'i'i3l
.02385
.99972
.04129
.99915
.05873
.99827
.07614
.99710188
28
.Chi'iiH'
.L'^L^nS'
.02414
.99971
.04159
.99913
.05902
.99826
.(»7648
.99706187
24
.Oiw;'js
inri'iSi
.02448
.99970
.04188
.99912
.05981
.99624
.07672
.99705186
25
.o<.fr-j:r
.inj;i',ir|
.02472
.99909
.04217
.99911
.05960
.99622
.OTTOl
.99708,85
26
.Oi>7r*t:
l^'O'ir,
.02501
.99969
.01246
.99910
.06989
.99821
.07730
.99701
84
27
.O-^v-.-^
.'r:,u.r
.02530
.99968
.04275
.99909
.06018
.99619
.07750
.99699
88
28
(yy^\ 1
■>:.■!■ ir
.02500
.99967
.04304
.99907
.06(M7
.99617
.07788
.99696
88
29
.©".k^n
/I'wS'
.02589
.99966
.04833
.999061
.06076
.998161
.07817
.99694
81
80
.0*K-^4:-S
.l'V'V<'i3
.02618
.99966
.04362
.99906
.06106
.99618
.07846
.99092
80
81
.©"Wi-j
Wy,^
.02647
.99966
.04891
.99904
.06184
.99812
.07875
.09689
29
82
.0<i'>:51
1 1'll: J'. 18
.02676
.99904
.04420
.99902
.06168
.99810
.07904
.99687
88
83
.OK'!".)!
■i-.i-.i-. 15
.08706
.99963
.04449
.99901
06192
.99806
.07988
.99685
27
34
.OKp-.^)
.!-<■.,■., 5
.02734
.99968
.04478
.99900
.06221
.09606
.07902
.09688
86
85
.'o:«M^
:^'i;m+5|
.02763
.99962
.04507
.99898
.06250
.09604
.07991
.99680
25
86
.OrcsT
I^'rn5
.02792
.99961
.04536
.99897
.06279
.99803
.06020
.99678
84
87
.Oi'.\-
'■■4
.02821
.99960
.04565
.99896
.06308
.99801
.08049
.99678
83
88
.01'
4
.02850
.99059
.04594
.998W
.06387
.99799
.08078
.99678
88
80
.Oil .,
41
.02879
.99959
.04623
.99893
.06866
.99797
.08107
.99671
21
40
.OUtH
:-i...r3|
.02906
.99058
.04663
.99602
.06896
.99795
.06186
.99668
80
41
.Oim^
1
.02938
.99967
.04682
.99890
.06424
.097%
.06165
.99666
19
42
.or;i^
wy.o^
.02967
.99956
.04711
.99889
1.06458
.9978^
.06194
.99664
18
43
.0 !-.':> 1
-.,■.,,2
.02996
.99955
.04740
.99888 ; .06482
.99790
.06223
.99661
17
44
.01:^^1
;,M-r3
.03025
.99954
.04769 1.998861 .06511
.99788
.08252
.99669
16
45
.OriM',
:*'.:v.ii 1
.03054
.99953
.04798
.9988511.06640
.99883; .06569
.99786
.06281
.99657
15
46
.01. i:^
I''!'.!-!! 1
.03083
.99952
.04827
.99784
.08810
.09664
14
47
.o-:ii^;
irrj'.ll ^
.08112
.99952
.04856
.99882 ll. 06598
.99782
.06389
.99668
18
48
.OM.:w>
/:n»^i'.iO
.08141
.99951
.04885
.9988111.06627
.99780
.06368
.99649
12
49
.01 [■,■.'
.kJ-j'.tO'
.08170
.99960
.04914 .99879!. 06656
.99778;
.06397
.99647
11
50
.OH,M
,0^j^J.i9:
.08199
.99949
I.04M3
.99878. '.00686
.99R78''. 08714
.99776,
.06426
.99644
10
51
.ohr;^
WKi-9
.08228
.99948
1.04972
.99774'
.06456
.99648
9
52
.oi.'.r!
l(M-9
.03"i57
.99947
.05001
.99875 1. 06743!. 99T72
.06484 .09688
8
53
.OI.M'I
l«LJ'i-^l
.03286
.99946
.06030
.99873
.06773
.99770,
.06518 .09687
7
54
.Ojrri
:j'k,i-Si
.08316
.99945
.05059
.99872
.06802
.997881
.06542 .99686
e
56
.(^■■■■-'
-^-n-7'
.03345
.99944
1.05088
.99870
.06831
.997661
.066n .99682
6
56
.C-;^-,-i
■4T 1-7'
.08374
. 99943 1 1.06117
.998691
.06860
.99764;
.06600 .99630
4
57
.Or<io'^
.vnj-i}
.03403
.999421 .05146
.99867
.06889L99762
.08620 .99627
8
58
59
.01 r,^:
.0l71ij
.03432
.03461
.99011; '.05175
.99940 1.05305
.998661
.069181.99760
.08668
.08887
.99685
8
1
.99WJ4, . .uD»<i<
.tfV<UO
.99688
60
/
.01715
8
.03490
CosiD
.99989
.06234
.998031 .06976
.99756
.08716
.99619
J
Bine 1
Cosln
Sine'
Cosin
Slnej
Oosiv
Sine
#
880 1
8^
r-
86* 1
85*
Digitized by CjOOQ IC
TABLES,
781
TABLE Vl.—Qmtinued,
Natural Sines and Cosines.
9
50
6«»
70 ,
g- 0-
"■"
Sine
Cosin
Sine Cosin
Sine
Cosin
Sine ICcKSin Sitie ICo';in
/
"0
.08716
.99619
.10453 .99452
.12187
.99255,
'.\mv;\ .990^ i . tvi4^ . mm 60
1
.08745
.99617
.10482 .904491
.12216
.99251
. lawe , . mm \ . i V5?.2 ^^ka . 59
2
.a8'r;4
.99614
.10511 .90446,
.12345
.99248!
. 131175 1 . mn i> 1 1 57ot . nfcrso 58
8
.08803
.99612 '10540 i. 99+43
.12274 .99244
AMm i .9'-«m:. ' . ihrm .VkQi^- 57
4
.06831
.99609 i!.ia569 .99440
.12:^1)2 .99240
.\Mm .mm ifiM-iH ^JHTsi' M
5
.08860
.99607,1.10)97 .9W.37
. 12:«1 1. 99237 ;.lJiN.[ M%m\ A^l^.mm »l
6
.08889
.996041,. 10626 .9»4M
.123(50 .99233
.l^i.KNi J|'.W2 ISSHI J»^4l 54
7
.08918
99602 .10655 .99431,
.12389 .99230
. 1 4 n u . flHlri H 1 . 1 rifti-j OHTar ' 58
8
.08947
.99599 .10684 .99428
.124181.99226
M\\^ y^wM ir^T;} gwiaa 52
0
.08976
.99596!
.10713 .99424'
.12447 .99222
. 1 f 1 77 .^fKhi , 1 tm^l \ .aSTiiS- 51
10
.09005 1. 99594'
.10742 .90421 1
.12476
.99219
.llJLO,.9f3ii6(ij JSOai 1.987^ 50
11
.00084 .99691 1
.107711.99418
.12504
.99215I
.113^1 flNttf^sl' \^W m\w \%
12
.09063 .99588!
.10800 .90415
.12533 .99211,
. \\2^\ ,i>str7H . ] r,y^ .»*714 1 48
18
.09002 .99586
.10829
.99412
.12562
.99206
. ij-jie . 08073 1 . 1 aoi7 i .bhtoo 47
14
.09121
.99583
.1068
.90409
.12591
.99904
.ir^'ip ssm>\^\ .lf)O40 .gttT04 46
15
.09150
.99580
.10887
.904061
.12620
.992001
.1(;W'J .tHMk') mFjLge^flO 46
16
.09179
.99578
.10916
.90402
.12649
.99197'
.ii»:b
,»^Wli .leiOiiM^iPS 44
17
.09208
.99575
.10945
.99399
.12678
.991931
.14*07
.S^lft^7 .lOlJJi' W^lJO 48
18
.09237
.99572
.10973
.99396
.12706
.99189
.11436
.w^:i^ .vMHv.m'm 42
19
.09266
.99570
.11003
.99393
.12735
.99186'
.11464
.9>^fM.^ Jf'r^l^ 9^^^^^1 41
20
.09295
.99507
.11081
.99890
.12764
.99182
.144^13 ^.&-l'J4 .iNiil^ iKM.nO 40
21
.09824
.99564
.11060
.99886
.12798
.90178
.line-j'.^MMi"! ii'.e^Jt r^f^Ti 89
22
.09353 .995621
.11089
.99383
.12822
.99175
M~''\ .Iir!*:/, iK.x'ITj ^-■ri'iT 88
28
.09382
.99559
.11118
.99380
.12851
.99171
M-^^^\ :^'^ .T^-iiii l»M.-.2 87
24
.09411
.99556
.11147
.99377
.12880
.99167
M^m JMs^, Jt^i .t^-i.:.; 86
25
.09440
.99553
.11176
.90374
.12908
.99163
.I4iw3r7ues8e8| 1 J03CI m:^ 86
28
.09469
.99551
.11205
.99370
.12937
.99160
.\\^m\.mm\ .leafii) -it^i.^^ 84
27
.09498
.99548
.11284
.903671
.12966
.99156
.14110,-5 '.WPhI 16411* *mA^ 88
28
.09527
.99545
.11283
.99384
.12995
.99152
.l.j:-J;V/M>Hi' .164471.1*^638' 88
29
.0a">56 .99542]
.11291
.99360
.1:3024
.99148
.M::c' .:j^1«'i. .UH70 .Wi633 81
80
.09585
.99^
.11320
.99857
.13058
.99144
.l-s:>*i ■ .U^wi . IGMS 1 . Liee2»| 80
81
.09614
.99587
.11849
.09854
.18061
.99141
M^v'^.^^mn
.m^'.9ft6Si'29
82
.09642
.99534
.11878
.90351!
.13110
.99187
.l-J^:i>^ Wim
Ki5(i3 .(NilO 28 1
88
.09671
.99531
.11407
.993471
.13189
.99183
M^i\.W!m
J65ftl
.Q@614' 27
84
.09700
.99528
.11436
.99344
.18168
.99129
.14HS0l.g8^
.ifjaao
.06600 26
86
.09729
.99526
.11465
.993411
.18197
.90125
.l4H!iSi.9«S^
-lesiB,
.96004
25
86
.09758
.99523
.11494
.99337
.13226
.99122
.14;«M .98870
.16677
.99600
24
87
.09787
.99520
.11523
.90*m
.13254
.99118
.i4r<ftf .sm;\
j67fle
.96096
28
88
.09616
.99517
.11552
.99:fil
.13283
.99114,
.15011 .88867
.16734
.oeeeo
22
89
.09645
.99514
.11580
.90'»7,
.13312
.99110'
.ifiiMo ."mm .1*376;^
.9^585
81
40
.09874
.99511
.11609
.99324
.18341
.99106
.i6<»8 .wm j.l<nvJ
.mm
20
42
.09903
.99508
.11688
.09820
.1»>70
.90102
.IROGT .W8M .IflflSO
.Q8G7B
19
42
.09982
.99500
.11667
.99817!
.13399
.99098
.itiigfl .e«B4B|,.nmo
.irjl55 .118845 Ll6878
.0«570
18
48
.09961
.99503
.11696
.99314
.13427
.99001
.^m^
17
44
.09990
.99600
.11725 .99310
.18456
.99091
. ir. 1 K4 fiNii 1 1 ' . 7 lVB|-« ■ . JrfLWl
16
45
.10019
.99497
.U754 -993071
.18486
.99087
.ir.-i- '--■' ^'^ ■• '^--.-^i 16
46
.10048
.99494
.11783 .99303
.18514
.99083
.ir.^'ll :- . ■ -i .- \ 14
47
.10077
.99491
.11812 .99300
.13548
.99079
.ir.-:i' '--:\ 'w^y. .^-r.Jrj 18
48
.10106
.99488
. 11*40
.99297
.13572
.99075
.ir.-i'i ^'^-^i^ ,17IJtWl!.ttrvi4r 12
49
.10185
.99485
'.11869
.99293
.13800
.99071
.ir-:!-',- :--iH .i.:^if<ii.ftH^i;}6| 11
60
.10164
.99482
.11898
.99290 1 .13629
.99067
.ItNiDG . L^SM I , . 17UCM . m^\ 10
51
.10192
.99479
.11927
.99286
.13658
.99063
.ir.3R5L?W80ft'i jncff .PeS36 9
62
.10221
.99476
.11«6
.99283
.136871.99059,
.lMHL9R?i)g -iriSfi OBTiSl 8
68
.10250
.99473
.11985
.99279
.13716 .900551
.v^xi .t»swo . 17164 LoaMei 7
64
.10279
.99470
.12014
.99276
.13744 .99051
.iMTi o'^Tw insaloflsiil 6
55
.10308
.99467
1.12043
.99272
.13773 .99047'
,V^m .^TTMi 1 ^^=£21.085061 5
56
.10337
.99464
.12071
.99269
.13802 .000431
.V<m .9«7K7 ATmuWm. 4
57
10366 .99401
1 .12100
.99265
.i:*«l 1.09039,
.\m- ,flMTH3' iTirro .wtiwsl 8
58
.108951.99458
.12129
.99262
.i:«60 1.90035
.lfj,wn .»*77S'MT308!.9(i4!*ll 2
59
.104^ .99455
.12158. 99258 ' .i;W9 .99031
.15615 .«Hr:^'. 17:136 .flWRfl; 1
1
.10453 .99452
Cosin , Sine
84*
.12187 .99255 1 .i:i917 .90027
Cosdn 1 Slue
.17:^i5 ,9mwi
_0
Cosin 'Sine
Cosin Sine '
cositi mm
#
83* :
82" 1 W
\ 80*
D
gitized b^
/Go
OS
782
SURVEYING,
TABLE Vl.^ConHnued,
Natural Sines and Cosines.
10»
Sine Coaip
1 ,j73^ .mm
4 ,17m .&8461
0, iTsa? .QfttSO
?|.175tiS .8^151
10I.17S5] ^y^aa
ll,Ll7B90l,tlSta5
la 1, 17737 .9W11
14 .i7Tria..M**oti
15 ■""- "■"■
Si tip 'Cosln
JIM '.7 ;H|4<;
19281
.loaou
.I933S
.mm
. 19452
J94@l
1050$
J»H118
A*Klia
,D93l07
Mm
,S837S'
.1785^1
.17B80
.17909
.1TU37
fil |j7«6e
IBS .ITim
g4i.lS05ti'.yf^t5-
g8MBl»KJ'.0H3aG|
SO MfllOTj .i^Si^i
,i8aa3',o8^o
JSI^I'.0^15
.1890(1 \IH31U
,ieffJM .iisjtji
.isao: aHeir.i
.i«ii»4 ,\ttt£5W.
» isHHi ,9rti>rr.
4i'.ia^Lti^7,
«'.ia5fi7',8«iKn
4a Mtiaosi„o8a5(j
44 IfiftiM .9«2:»o
45 .iwm'.9tei>
4e iHd^i urtsia
47 IHTIO .»*£H
48 .l»<MiHL»!tm»
49 .16707 .HWitri
ISTW ,9*;i«,
,11*538 .I'si^r;]
.I'.r-- • 17
A-x i:
.ILJ'. ■■ Yy
A^^'i\ 'J^i^jd
.lOfiSwI.fNhU
.IfC'J'i :iH:ilf}
Ji^ :. 'i-'j:*
Au;-.' 'i-r
jti;- ■ •][
,1IJ' ■ IM
. 1 1.^-^ I ' I
,l1l^■ ■• 'It
.lll'i.'- ;i -'i^
.18905 l.yrif'^
.300-32
,w:^K^i
12°
Sine iCogin
:2b79li. 97815
.208a)
.20848
.20877
.20905
.20988
.21019
.21047
.21070
.21104
.21182
.21181
.21189
.21218
.21246
.21275
.21803
.21331
.21360
,:;A.Ji:k;..ui"Lt.'ii!
.2t»l!Ji ,tJ7",>W
.21417
.21445
.21474
.21502
.21530
.21559
.21587
.21016
.21644
.21672
.21701
.21729
.21758
.21786
.21814
1.21843
I .21871
1.21899
.21928
13
.21956
I .21985
1.22013
1.22041
1.22070
CO
51 1883-1
53 .18«ei
N tA^m .^Wy
B5 11*338 .0^130
B6 , A^mi .iWlS-i
67 . l8tHiS .0fll7&
.^A)i."ii)|.^i>H7
.98312,1
.983a)7,|
1Q0S4 .y8ir4
10053 .lKil68'
0t»L 19081 ,,6816^
Cositi| Btiiti
.3m35|
.*)oa>
,200-1^
.Uf)?577
.:ftCl4
.t»7t[a
aoTflii
dofidnf
.oTsda
. E»7K57
.^7H15
W7M39
.ft7Hd7
A»7>»ai
,f:^l5
IS*
r
14*
"i!iM95'.af7437
,S£i^r^J'. 97430
.!£r/^2 .074'i4
.5-:>si> t<71l7
,:*L'i-»iiH L>;-iii '
.^\:;x: y74i.^
St7iSJ,.07;iS4 I
2-J7ri() .071S78J
.S&77B OTSTlj
,2->3?W .073&5
.Si.^%'vLflT3Se
a*^1 ' .tt73Sl f
,*^»jsi lP7:l:^8
.52is.NH .'.yrswi'
ife-uTT .grass;
.*2^itmG UT^iiyl
. 3^002, jjr:J04
■i;'JG4i
:C^7
l*rJ51
U?^44
.97237
60
^a CoBin
.»tl9« BTOSO ^
.»l2a> .iin>23 50
.34t41>|.W0I5: 58
^Mrr. Ar:\m 57
.:il3:kf. tiTOOl, 66
.1^^^ ,UGU»1 56
.atlfial.OOflS? 64
..^390 L^*^^ 68
.5W41m| 9t!SI73', 5«
.aiHn,{K»L«6 61
.^174 9<Jfl5Q;50
Bine
.97660
.97553
.97547
.97541
.97534
.97528
.221261.97521
.22155 .97515 I
. 22183 j. 97508 j
.22212 .97502 I
.22240 '.97496 I
.222681.97489 I
.222971.97483 I
i. 28825,. 97476 I
1.22353 .97470
1.22382 .97463
1.224101.97457
i .22438 .97450
j. 22467 j. 97444
I .22495 .97487
' Ck>sin I Sine
.2S:f73,
,143401'
.s£^oa7
.mm
.2i!n:i
.mm
.wm
.»4T56
,^71*1
34Hnl
.atH07i
.»4835
.mos4
,iS03H
9tlOSS'49
,3»ra»45l48
MW7I 47
OtHOIOi 46
9tit^i 45
W0t6t44
48
42
,41
.90887 40
97280
9?2^
oreiT
97308
,97109
. 97180
.071?^
iJ7l76
.97169
J>TM8,.
.2il7I'-J J)TM8
.3S740 .97141,
,^TUll],9;-ia4h
.2^j797 .97127 ,
.*^^2.^!.97iai
-;--■' 117113"
-• •■ '.iriOQ '
,95066
.£>0»l
.85122
.25161
.25179
.^1007
.8f5i?35
.asi>9i
.2r>stfi
.t£:>;j7e
.ir>460
.25188
.25516
.35545
.25^73
89
Kfe^TS; 88
1»:866^ 87
9f!85§ 86
,9(^1; 85
mnu; 84
9t;x37 88
9tl*^ 82
9t-*K^ 81
Uti«15 80
.90807
Mm
9677« 26
.fi677l!a4
/*-r^ 88
l<n40 21
.[Jti.742^90
.36734 19
967*7, 18
.Pi7l» 17
,9C712 16
9Ca5 15
'jmn 14
.yoijflo 18
mm 12
.\myr^ ii
t«t«7 10
.21023
.24051
.24079
.24108
.24136
.24164
.24192
Ck>sin
,97079
,970ra
.97066
,97058
97051
.97044
,97087
.97090
760
- \\
.Z''-:' ■H...15
.257ia ,9608
,25741 .HS680
.25709 ,gafl«8
.2&7g8 .96615
.85886 .90606,
.85664 .90600'
Cosin Sine
75^
Digitized by CjOOQ IC
TABLES,
783
TABLE Vl.—ConHnued,
Natural Sines and Cosines.
I60
Sine Ooein
25910
25966
259»1
,26079
26107
26135
26191
.26219
26247
28275
26806
26881
,26850
26887
26415
26448
.26471
.26600
.26666
.26612
.26640
.26668
.26606
.26794
.26762
.2<y780
.26806
.86864
.86820
.26048
.86076
.27082
.27060
.27068
.27116
.87144
.83172
.87200
.87228
.2?2S6
.27261
.87812
.27340
.87368
.27896
.87424
.87452
.87480
.27508
.27586
.27564
Oosin
.96678
.96670
.96562
.96565
.96^47
.96640
.96682
.96517
.96600
.06502
.06404
.96486
.96479
.96471
.96468
.96456
.96448
.96440
le^
Sfho
f'l-.sin
.^::^iA
.■"tw86
.8j:-.*^j
.■r,ii8
.87n;^J
.tit^llO
.2:ffl8
.'>il,(B
.Vf:\u\,
.-mm
.2::'H
.rrHfiC
.277:11
lN.i<)78
.Z.\r,'>
\u\m
.2^:^?
."Jtkl62
.2^Hi5
AWaM
.2:w:i
.*^mi
.27<'
-:h-^187
.2-.
e9
.2^
ei
lift
17-
18«
Sine
.29237
96425
964171
96410'
96402
96394
96379
96371
96863|
96855
96347
96310
06316
06306
06801
.06877
.96260
.96261
.96253
.96246
96222
.96198
.96190
,96182
96174
,06166
.06158
.06150
.06142
.06134
.06126
Sine
740
.2tkNM
.2KtM»5
.2Hr^J
.2e-t:s
.»
.21
.21
.21
.21
.21
.a
.21^
.a
.a
.2e
.21': •■•
.2h.-!
.2h-'''
.21^
.2J
.8h
.21
.2h:iu
.2H;i'-i
.2H^^.i
.2B^iV
.2H>J7n
.a
.a
.2h'''-.
.21'. M-.
.2!«'-'»
.21NV.H
.2!'1','<^
.2!'ir^i
.2l'-jO
^•■:-
Cc
05
197
,:.eo
.ii:i'«l
.lp:.',I72
.o::*J64
.iir,'.i66
i'7.48
40
81
33
115
107
00
«2
r74
165
157
^0
M.S41
fi2
«1
tlG
107
IW
«91
B2
.^■>,74
Sf5766
.lJ:ir40
.10740
-^'•782
■24
15
07
'-'08
i ©0
■>^«1
.ir^t73
fiM47
'*.'«9
ao
730
.29293
.29321
.29370
.20104
.29460
.29487
.20515
.20548
.20571
.20500
.20710
.29787
.29708
.20849
.29876
.29904
.29960
.89967
.30015
.80043
.80071
.80008
.80126
.80154
.80188
.80200
.80287
.80266
.80820
.80848
.80676
Ccxinj
.tt
.a
.ft
.9
.91
.»
.9i
.»
.0
.0
.»
< '5
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.9
.9
.9
Sine
.30902
0
9
9
0
0
0
.0
.0
.0
.0
.0
.0
.0
.9
.0
.0
.0
.0
.0
.0
.9
.0
.9
.9
.0
.9
.»
.0
.»
.9r
.»
.91
.9
.91
.91 "
.9!-l'
Sin
.30957
.30985
.31012
.81040
.81068
.31095
.81123
.31151
.81178
.81206
.31233,
.81261
.312891
.31316'
.81314'
.81372
.81800
.81427
.81464
.81482
.81510
.31537
.31565
.81593
.81620
.81648
.81675
.81708
.81780
.81758
.81786
.81813
.81841
.81868
.81806
.31023
.81051
.81070
OobJii|
.05100
.Ofeogr,
.fl5c»|BS
.05(rr9
.O&K^I
.06<>4a
ot^ra
.OfH>J-J
,ot^n
■!
.82064
.82061
.82060
.82116
.82144
.82171
.82109
.82227
72* I
452
.82800
.82864
.82419
.82447
.82474
.32557
Cosin
.9ffH>i
.94'^',i:
.94^4^H^
.9^'i:o
.94';KU
.041):^ J
.94ma
.94i»;i3
M\U\
.9^>i€
.0IMJ7
.9<iSK?J
.9aH78
.Os%^
.Oisijo
.9^|!H42
.Ot?^.^-*^
.OlHU
.0IXJI5
.017^5
.01 Xj
.017:7
.0i:w
.Oi7'^
.917'i0
.0.17JO
.OITl^J
.0^71 '3
.OSTil-J
.Oam")
.O^riisj
.Wi;74
.O^H.W.'J
.O^iiTiS
.046 id
.OltiST
o*(i2r
.94*5IS
94JVKJ
94- :j^, ;^
Oi:.i»U .;i:
OJL^iHJj..**!^
ir
Sine
Oosin
.33587
!U\^fi
.asi^
:m.M2
.aaeia
'.M-.«
.a2ti;iv»
.fn:.-J3
.aL'^tfi-
Liir44
.aii^
mr^il
.ajT^h!
.uiiie
,3ir7l!l
.yi4S5
.3^>rrT
':iur6
.r^:: 01
■Ullfi
•''-"-■ ■ ^
'■ii-»7
.3;i(E/il
.3:}<>7y
mm
.3:sm
.3:1:. J.-,
,Si:.7:i
.3?iNl&
S90SJ>
3<10!T
.94571
.94561
Sii^e
I .8414^
.54175
Coaiji
iui7
IU-.J8
'.1!I8
10
10
it-., -JO
ui^iHO
1W370
.'M3(J1
M351
UI342
^n'J2
I3i:il3
.W.^H4
LH-r4
.^^i'i14
■.I Vl^
Mm
,U^UU7
,&4157
,Wli7
,L4i:-{7
in 1 18
Mm
.yjiif«
'sH^'(J8
-.*hi40
0
0
.;.:.-<0
.imso
. my^'Jd
I^l_iUJlL2i!iiigle
7^4
SURVEYING,
TABLE VV^ConHnued,
Natural Sines and Cosines.
9
20« i| 21» (1 22'* II 230 I
240
^
Sine
Cosin j Sine Cosin!
.93969 1.85837 .ft3358;
Sine Cosin 1
Sine 1 Cosin
Sine Cosin
"o
.34302
.374611.92718'
.39073,. 92050
.40674 .91355 60
1
.34229
. 93959 !..35Hfr4,. 93348,
.37488 .927071
.39100 .92039
.40700 .91813 59
2
.34257
.93M9
.85891 .93337
.376151.926971
.39127'. 92028
.40727 .91331 58
8
.34284
.93939
.35918 .93327
.37542 .92686 11.39153 1.920161
.40753 .91319 5?
4
.34311
.93929
.a5945 .93316
.37569;. 92675, .39180. 920051
.40780 .913071 56
6
.34389
.93919
.859731.93306
.37595 .926^4 I.SftW 1.919941
.37622 .92653 .39234 '.91982
.40806 .91296 55
6
.84366
.93909
.86000 .98295
.40833 .91283
5C
7
.34393
.938091
.36027. 93285
.37649 .92642 1 .39260 .91971 1.40860 .91272'
53
8
.34421
.93889
.86054 .93274
.37676 .92681 | .39287 1.91959 1
.40886 .91260
52
9
.81448
.93879
.860811.93264
.37703
.92(^.'. 39314 1.91948,
.40913 1. 91248
51
10
.34475
.98869
.36108. 93253
.37730
. 92609 ;|. 39311
.91936
.40939;. 91236
50
11
.^1503
.93859
.86135 .93243
.37757
.92598
.80867
.91925
.40966
.91294
49
12
.34530
.98849
.36162 .93232
.377tW .925871 .89394
.91914
.40992
.91212
48
13
.34557
.93889
.36190 .93222
.8781 11.92570 I. 39421
.91902
.41019
.91200; 47
14
.84584
.9aS29
.86217 .93211
.87838
.92565
.925M
.89448
.91891
.41045
.91188 46
15
.31612
.93819
.36244
.93201
.37865
.39474
.918791
.41072
.91176 45
16
.84639
.93809
.86271
.93190
.87892
.92543
.895011.91868'
.41098
.91164144
17
.34666
.93799
.86298
.93180
.87919
.92582
,.395281.91856,
.41125
.91152 43
18
.34694
.937891
.86325
.93169
.87946
.92521
.395551.918451
.41151
.91140 42
19
.34721
.93779
.86852
.93159
XCiiri .92510
'.395811.91833,
.4117«
.911281 41
20
.84748
.93769
.86379
.98148
.87999 .92499
.39608
.91822 1
.41204
.91116.40
21
.34775
.93759'
.86406
.98187
.88026 .92488
'.89635
.918101
.41281
.91104' 39
22
.34803
.93748
.36434
.93127
.88053 .92477
.89661
.91799
.41257
.91092 38
23
.348301.93738'
.3ft461
.9311G
.86080 .92466
.89688
.917871
.41284
.91060 37
24
.34857
.93728
.8^488
.93106
.38107 .92455
.89715
.91776'
.41810
.91068 36
25
.84884
.93718
.36515
.93095
.381*4 .92444
.89741
.917W
.41837
.91056 35
26
.84912
.937081 .86542
.930&4
1 .38161 .92432
.89768
.91752
.41863
.91044 84
27
.84989
.93698, .36569
.93074
1 .38188 .92421
.89795
.91741
.41890
.91032 88
28
.84966
.93688 '86596
.93063
1 .88215 .92410
.89822
.91729
.41416
.91080; 82
29
.34993
.93077 .36623
.93052
.:-8241 1.92399
.8984^
.91718
.41448
.91006 81
80
.85021
.93667
.86650
.98012
1.88268 1.92388
.89875
.91706
.41469
.90996180
81
.85048
.9365?
.88677
.93081
.882951.92877
.89902
.91694
.41496
.90064 29
^
.35075
.93ftl7
.36704
.98020
.88822,. 92366
.39928
.91688
.41522
.90972:28
83
.85102
.93637
.36731
.93010 .88349 1.923551. 39955
.91671
.41549
.90960 27
84
.85180
.93626!
.86758
.92999' .883701 .923431 .39982
.92988 ' .aW03,. 92382 .40008
.91660
.41675
.90948 26
85
.86157
.93616
.86785
.91648
.41602
.90686,25
86
.85184
.93606
.88812.92978 .88430
.92821
.40085
.91686
.41628
.90&M 34
87
.85211
93596
.86889 .92967 .38456
.92810
.40062
.91625
.41666
.90011 23
88
.85239
.93585
.868671.92950 .88483
.92299
.40068
.91618
.41681
.90699 22
89
.85266
.93575
.36894 .92045 . .38510
.92287
.40115
.91601
.41707
.90887 21
40
.86298
98565
.869211.92935
.88687
.92276
.40141
.91690
.41784
.90675120
41
.85820
.93555'
.86948 .92924
.88664
.92265
.40168
.91578
.41760
.90663*19
42
.86347
.93514
.36975 .92913
.88591
.92254
.40195
.91566
.41787
.90661 {18
48
.85875 .93534
.87002 .92902
.88617
.92243
.40221
.91555
.41818
.90639' 17
14
.854021.93524
.87029 .92892
.38644
.922;}!
.40248
.91543
.41840
.90626 16
45
.86429L93514
.87056 .92881
.88671
.92220
.40275
.91531
.41866
.90614; 15
46
.85456 .93503
.87083 '92870
.88698
.92209
.40301
.91519
.41892
.906021 14
47
.854841.93496
.871 10 1.^2859,
.887^25 .92198
.40328
.91.508
.41919
.90790 18
48
.855111.93483
.87137 .928491
.88752 .92186
.40355
.91496
.41945
.90778
12
49
.85538 .934721
.87164
.92838
.88778 .92175
.40381
.914»1
.41972
.90766
11
50
.85565!. 93462*
.87101
.9e827
.88805
.92164
.40408 ■.914?2
.41996
.90758
10
61
.85692'. 93452
.37^8
.92816
.88832
.92152
.40484 .91461
.42004
.90741
9
62
.35619,. a3441
.37245
.92805
.3?^9
.92141
.40461 .91449
.42061
.90789
8
53
.35647!. 93-131
.3?272
.927Wi
.38886 .92130 t
.40488 .91437
.42077
.90n7
7
fA
.35674 .93420
.3?299
.92784 1.889121. 921191
.40514. 91425
.42104
JB0704
6
65
.857011.93410 ' .3r32<J
.92773' .38939 .92107
.405411.91414
.42180
.90692
5
66
. 357^ .93400 .37353
92762. .mm .92096
.40567 .91402 1.42166
.90680
4
67
.35755 .93389 1 .37380
.92751 1 .38993 .92085 ' .40594 .91390 .42188
.90668
S
68
.35782 .93379 .37407
.92740' .39020] .9207:3 .40621 1 .9137«
.42209
.90666
2
59
.35810,. 93368 1.37434
.92729 .39046 ,920621 .40647. 91 866
1.42285
.90048
1
60
.358371.93358 1.37461
CosiQ 1 Sine { Cosin
.92718
Sine
.39073 .92050 .40674 .91355!
i.42S62
.90681
_0
Oosin
6"
Sine j Cosin Sine
Cosin
Sine
69» 1 68« 1
^- il 66« Di,J
zed b>^feO(
TABLES,
785
TABLE Vl,^ConHnued.
Natural Sines and Cosines.
0
S«^
1 20^
_E7' 1
1 ^®' '
29«
60
SinP CoslQ
Sine
wyim
' Sine
Cosln
Sine ICosin
:4H-:^T
.I^TU
.46947
.88295
.484811.87462
1
..{'.»->-, in HUH
,4.H^i^1
R^ts*;-
.i-rta-J.^JtlHTl
.46973
.88281
.48506 .87448
50
2
.;"■•„■- ■ >»■-,
.]:■ '
vi.-:^
,.4%r.r K^Mivi
.46999
.88267
.48582
.87434
58
8
. J-
■^41
.4^177 .>'-"4
.47024
.88254
.48557
.87420
67
4
,-. ;■
.]
-.-H
.4v.jici >.**itH;
.47050
.88240
.48588
.87406
56
5
.VSVM
> '"ili>
_ j:
'>1S
.4.^^-^
.?:<i«i;j:}.
.47076
.88226
.48608 .87891
56
6
.49t2»Jl/N)r-r
^■^i3
.4.*'^^4
.SKREl
.47101
.88213
.48684
.87377
54
7
.42445: TK^%i5
A
-rm
.4,i5H0
.mm
.47127
.881991
.48659
.87863
58
8
.mT^
■iir^'v^
1 .d-l'H^i
.H!rTT7
.4V/1C
Mm^
.47153
.881851
.48684
.87349
52
9
A'im
,!Wj3n
.44i>TJ
.ftyrf^»4
.4V5)t>
.^m^\
.47178
.88172
.48710 .87835
61
10
,^as45
.0(JuO7
.41W^
.8W7a5J
.4m^
,mm\
.47204
.88158,
.48735
.87821
60
11
.43552
.m^
A\n\
.«173^^
.4r.^>M
...,,^l
.47229
.88144'
.48761
.87806
4d
12
.4sm
,flOl83
.44Tr-l
..S9^2G
.4S:iii - '
.47255
.88130
.48786
.87292
48
18
.4mi
.^MHTD
.4-4177
J5:ms
.45756
.^^-.■-s
.47^1
.88117
.48811
.87278
47
14
.42991
jfliir^s
.4!'3'Jli
.89700,
.407^
,6t^l5
1.47306
.88103
.48837
.87264
46
15
.*S667
. w 1 Hi
.4!e-J0
,8068TJ
.45787
.aeons 1
1.47882
.88089
.48862
.87250
45
16
.-lasa
.viM^ti
AXl^^
.ao«74i
.46813'
.47358
.88075
48888
.87235
44
17
.4-i:TOi
,1hi|C]
AV^l
.896^3,
A^^m .-- .
.47383
.88062
!48918
.87221
43
18
.-l'>7Bi>
.1«i|jN
Avm
.8064a'
.4fV'«05
.47409
.88048
.48938
.87207
42
19
Ai'^rvi
.!Hi:-k;
.4im
,mm
.Ci«l-t
/- - - • '^
.47434
.880a4
.48964
.87193
41
SO , .-i*;;.-^-
.:.Mi;;^i
.44350
.mm
.43^17
.47460
.88020
.48988
.87178
40
21
.-t->2r»
.fNVS7l
.4^im'5
.w&m
.*^2
.47486
.88006
.49014
.87164
89
22
Am^i
.lftifi-*s
.-t - ■ ^-"loflr
.4.VJfJJ^
.^
.47511
.87993
.49040
.87150
38
28
,4sm
.9tKWJi
. 1-
..-.^
A^/m
.'■ . 'r*
.47537
.87979
.49065
.8n86
37
24
jm^
.Oimi
'a '
■ i:.7i
.400-31
.>-. -i
.47562
.87965
.49090
.87121
36
25
.43flS»
.txmt
AW^*
.^a-*^
.*KHr»
Ibxl'-^
.47588
.87951
.49116
.87107
36
26
,mm
.tKir«0
AVA<\
.Hii^iA
,*^I?J
.8:i;:4)
.47614
.87937
.49141
.87098
84
27
.4sem
.,9(>irw
Av^i'l
.H^+.:itr.j
.^-ifLtr
..■SSTM
.47639
.87923
.49166
.8'.'079
83
28
A^^m
.lhf:^l
.4r.^.>
>im%
.^ilL^l
.J^7:.t^
.47665
.87909
.49192
.87064
32
29
.iSi^r*
j,j i-n
Ai:>\y\
.M^P.'Vlfl
.#iito
.8M:irj
.47690
.87896
.49217
.87050
31
80
A^^l
.SW«5a
,4'iWu
.b'>ii^
.4<;31T5
.8tt:oi
.47716
.87882
.49242
.87036
30
81
.mrr
.00346
.*1F5HJ
.8^%
.46301
,80608
' .47741
.87888
.49268
.87021
29
82
.4,^!0l
.90-J33
.41tnJ
Mmi
.46^*6
.88(iT4
.47767
.87854
.49293
.87007
28
83
.4^130
.iKKai
.41fJVi^
JilH54
.*J2GS
.88fi(}l
.47793
.87840
.49318
.86993
27
84;
A^iao
.^3208
.44?^i
,WMl'
.4f?i78
.88617
.47818 1 .87826
.49644
.86978
26
85
.4ai8^
.00196
,44750
,8at:JS>
.4^304
.mm
.47844
.87812
.49369
.86964
25
86
,4ssm
.fWlSI
.^irre
.^IS
,4Aiao
.88ceo
.47869
.87798
.49394
.86949
24
87
,43335
.1K)171
,4490^
M¥^
.4fi3S3
.886071
.47895
.87784
.49419
.86935
23
88
.43SflI
.90158
.44838
.mm
,46^581
.888^^8
.47920
.87770
.49446
.86921
22
89
.43387
.Oi:H46
.448&*
.80876
.*)tn7
.mmi\
.47946
.87756
!. 49470
.86906
21
40
.43313
.00133
.lifltM
.iy^eo
,4t;43!J
.885<jfl
.47971
.87743
.49495
.86892
20 *
41
.4^0
.wm
,«90^
.s^Qsn
' AFA7,H
.mm^
.47997
.87729
.49621
.86878
19
42
.*jattij
M\^
.41 ■ • . ,
.IIVISI
,m^^
.48022,. 87715
.49546
.86868
18
48
.4.'mJ
.^\m\
.4XilO
.SSlySGi
.480481.87701
1.49671
.86849
17
44
.4.*iia
.m^m\
A-' ' J
.4*5536
.8?t512'
.48073
.87087
1.49696
.86884
16
45
.4'^ri
,4MH'J
.^j^j^
.#j5C1
.S*1L«
.48099
.87678
1.49622
.86820
16
46
,jmn
.4''ri)^lfl
.fm^
.4^587
.S6M«5|
'.48124
.87659
1.49647
.86805
14
47
.■Ui497
.mm\
.^-^^itw
msn
.40613
.^IT'i'
1.48150
.87645
1.49672
.86791
18
48
,t3S3S
.mm
.45t)8iS
.e93S9
, .46^30
.eBirj^i
, .48175
.87631
,.49697
.86777
12
49
. fa&4iJ ,Of>HO
.411 14
.^ie^m
'.«}»KH
.»I4I5'
'.48201
.87617
!. 49728
.86762
11
60
.-13676
.OCUOT
.4^^140
.88^,
4ficyo
J^itaii
.48226
.87603
1.49748
.86748
10
51
.43002
'.TOW'
.4fitl3fl
/BOSlfi'
.4^'71^'.
. •■:'
.48252
.87589
.49773
.86788
9
52
.4^m
.HUl-JSl 1
.4f.iaa
,^i(Kl
.407 n:
;
.48277
.87575
.49798
.86719
8
53
.4ia;M
tiNii^
.^'il'^-
^".nits
' .^^TGT
..-.--..nj.
,.48308
.87561
,.49624
.86704
7
64
,]::r;-ji
-■■-■H-.l
a:'-^'-
.^■■M.**)
, .467113
.t*l:t77|
1.48328
.87546
.49849
.86690
6
66
.-i::";iii.
-i*!-5
.-I,.-:' v.«]t?7
.415819
.8H3i;3'
.48354
.87532
.49874
.86675
6
66
- j-i^i'
4. ■ ■►j^a
. .4*3844
.Bft^^lU
.48879
.87518
1.49899
.86661
4
67
■h-l
.1 MiO
.4IJ870
.8f«3«
.48406
.87504
.49924
.86646
8
68
,. I-.
.] MiJ7
,4^i™o
.mv^
.48430
.87490
.49960
.86632
2
60
.4u,.., .n^(JH
,4^5^31
.8a^m
.48456 '.87476
.49975
.86617
1
00
\ ^:K5■! 'ttU!H:S
.4."i;i9»^.Wl*vi
.4«S.*17
.?^J!>5. .4^4811.87402
1.50000
.86608
0
/
tMt.li; mn©
CosUj \ Sine
€(Jia'!j i SI u e Cosin j "Sln©
1 Cosin
"Sine
9
w
83"
es- ll 6P
1 W
3()
>gl^
786
SURVEYING.
TABLE Vh^Continued,
Natural Sines and Cosines.
30*
.imm
.501 TO
,50001
.fiozrr
.man
.50»S7
.503fl:ir
.B0377
.50«Da
.fiOG03
.5UIJ2S
.50771)
.50804
81«
.5067^1
.50«0i
.rj<KJ79
MiU
.51*2^ >
.51379
.M479
.8(151)1
.8Wi;i
,@6S10
.mm
. si:^IT
mi^v
.8fil'iJ
.will n I
.Km\
. K*hK!M| '
.B5a^.i
.K^X 1
. K57!^J '
„8:iT-i7i
.Ri7l7'
Slnu I
jSIne
.61504
.51529
.51554
.51579
.51604
.51888
.51653
.51678
.51703
.51728
.61753
.61778
.61803
.618S8
.61868
.61877
.61902
.61927
.61952
.61977
.60002
6r
.52051
.52076
.68101
.68126
.62151
.62175
.62225
.62260
.52275
.62299
.62349
.62374
.62423
.62448
.65Mr3
.6fU96
,85325
.58547
.5257^
.62597
.52646
.62071
.52720
.62745
.52770
.52794
.52819
.52»44
.62918
.52943
.52967
^52992
Cofiin
.85310
.85294
.85279
.86864
.86849
.85234
.85218
.85203
.85188
.851;^
.85157
.86142
.85127
.85112
.86096
.85081
.SK)66
.85051,
.850351
.85020|
.85005
.ft4989'
.84974
.1^959 1
.84943
.ftl913l
.848971
.^1882
.&4866
.84836'
.84820
.84805
Sine >
68»
82*
Sine ICoBin '
.62992 i. 84805!
.630171.84789
_ S3*
81 rn? Cfisln
W^
.53041
.63066
.63091
.53116
.68140
.68164
.68189
.63214
.63312
.68337
.63361
.63411
.63485
.63460
.68184
.58500
.63534
.63558
.63583
.53607
.53056
.53681
.63705
.53730
.5377M
.537r9
.53804
.53853
.63877
.63902
.63951
.68975
.64000
.64024
.54049
.54073
.54097
.64122
.64146
.54171
.54195
.84774
.84759
.84743
.84728
.84712
.ftl697
.84681
.»4666
.84660]
.846851
.84619
.84604
.84588
.84573
.84567
.84542:
.84526
.84511
.84405
.84480
.&14&4
.&1448
.g4433
.84417
.84402
.84386
.a4370
.&4.355
.&4339
.54 iS4
.64.M;^
.54.vi7
.64:^*1
.51. '--1:1
.5H'lii
.54^^*:.
.54
.5
.54 ■-■
.547T|iJ
.5t7:>(i
.64r^t
.54'iiW
.5
.5
!64yoi : imi^ j
.64n'.nJ .Ki'-flT
.K-Hhji.KlVit
.64844
.54269
.54293
.64317
.54342
.54366
.54391
.54415
.54440
.544^
Cosin
.&4308
.84292
.»1277,
.842611
.84245
.&i230
.84814
.84198
.84182,
.84167
.84161
.&4135
.841201
.t»104|
.84088,
.WO72!
.840571
.84041;
.84085^
.840001
.83994
.83978!
.5tHM.s
.H-i(-<r,
.6f.^i, r
.>- iSi.'l
.6l-''->:
. '^■'li'-^',
.Kmj:
^ ' ;: .;
.6f'i:
.K.p
.83946'
.838K3
.83867
Sine 1
6r^ii!^'.e!Err3
.K ,".11 =8.-1331
.6{-iSl .NTilUfj
.6;hV:-i;i .rtriioa
.K.r.:u .«;^5147
.5'.-^l jyjlJil
.65005 .83115
.66630 1.83098
.55654
.55678
.65702
.65726
.5'5730
.55775
.55799
.5VS23
..W847
.55871
.55895
.55919
Cosin
.83060
.83084
.88017
.83001
Sine ICodnl
".visiiv ".8^.ii>4lea
.55111a .K!>H7'69
[KVifiH .H'>71'. 68
.CM'J'ri .»t>ri6 W
.:M:HiMi f^'^td M
..■nHi4>f s-^'i2i 55
:rfi<N;( >.!isj.l6.54
.:^'.iK^ H-riK)' 58
./>4ii:ii^ ^2;;S715.
.:.'■!-■ • J4'4»
» 48
.bi.r.:^^'^ -.-n|8' 47
,&S\3&<.V.j«:.f^75' 46
,Wi?3HTi. ^■jc.-j© 45
.,...,.. ....j3 ^
S6 48
JO 48
.;«->- r-^.;.:B'41
.5W01 1. 82577 j 40
■ T' - 14 38
» 37
• ■ll'3«
.r^^-:i ^JsJ6 98
.r>'-.ir> s-jjm 34
.;,.;.■> -:i.l8 38
16 88
» 31
.. .ii -as 30
.s<k:«M Nsrt» 28
.5eT^i.8£SH7 26
.,VlT™i'.t3vr*^» 25
.r>r-'l -J- 14 '84
>7 88
n 28
k 81
i8|ao
.r.r tt'lO
14 18
.1'- ■ e, 17
.r...-,- -.:-tt'16
..■^u\&' 15
>^]48 14
tavs IS
.88115 19
.88008 U
.88088 10
.82904
Sine I
57«
66*
.fjTiii.iii>
.670111
.67071
.67005
.67119
.9n48
.67167
.67191
.67815
.67888
.67888
.67886
.87810
.57884
.67868
Ck>&in
.88066' 9
.880481 8
.880881 7
.88016 6
.81900 5
.8198^1 4
.81966, 8
.810491 9
.81088, 1
.81915 O
Sine
65«
!fe
'J'ABLt,i>,
/o/
TABLE Vl.^Continued.
Natural Sines and Cosines.
35*
86* 1
ZV 11 38--
89- ^
/
_fiiiie L\»siri
Sine
Ck>8in
Sine ICkMdn'l Sin^^
{'tv^hi
Sine 1 Cosin
"0
,tuX^ ,M]:ii5
.58779
.80902
.60182
.79864 |.6i:'iv.
\ ^>! 1 [
' ■1321 75715 ;«
1
.67;-^1 .H]y!^^
.58802
.80885
.60205
.79846
.6
- . ^ •:
' vfi5|.77696 60
2
.5741 *:> .S1S.H2
.58826
.80867
.60228
.79829
.6
177 .77678158
8
.b:i;iJi .H]H>o
.58849
.80850
.60251
.79811
.6
' 00,. 77660 57
4
^H\ri^ .^1H-;S
.58878
.80888
.60274
.79793
.6
62 .77641 56
5
.riTiTT ..Kirti'ij
.58806
.80816
.60296
.79776'
.6
45i.77623!65
8
.f;:-rfH KiNrS
.58920
.80799
.60821
.79768
.6
«8 .7760.5; 54
7
.r ?
.58948
.80782
.60344
.79741
.6
190 .77580 58
8
.12
.58967
.80765
.60367
.79723
.6
18;. 77568 58
0
.1 5
.58990
.80748
.60390
.79706
.6-. -
;85
.77560 51
10
.1 3
.59014
.80780
.60414
.79688
.617^^5
^7Mliii2|
,t;;ii68
.77581
50
11
.1 - I
.60037
.80718
.60487
.79671
.eisie
.78aM'
.6.^180
.77613
40
12
.t., I.J.I m:, 11
.59061
.80696
.60460
.79653
.t\M\
.m^s
.fi^i:308
.77494
48
18
.nrhfiT .S]f;'KS
.690ft4
.80679
.60488
.79635
.01 ^.n
.TS&68
jJ.3iS6 1.77478
47
14
.rCi-'.il .Slf^:l
.69106
.80662
.60606
.79618
.ejw<
.7^-^
r?j48 .77458
46
15
.f::- •■ v|
.69131
.80644
.60529
.79600
.6
.7"'."''
■ :r71 1.77489
45
16
.1 r
.69154
.80627
.60558
.79583
.6
>93 .77421
44
17
.1 . I
.59178
.80610
.60576
.79665
.6
[16|.T7402
48
18
.{'-.\^n :-M.)l
.69201
.80593
.60509
.79547
.6 K-
, ^ , ;. ■
E38;. 77884
42
19
.5:^10 >.i,vjr
.69225
.805761
.60622
.79580
.6-'l
,' " '1
«1 .77866
41
SO
.CTivii^.^]:.^
.69^
.80558
.60646
.79612
.6v-J
. ■; - i 1 -■
(88.77847
40
21
Xu^^^'.H'm
.69272
.80641 1
.60668
.79494
l.fl
m .WSA
89
22
.^r^s.H| >[-.!«
.59296
.80524
.60691
.79477
.6
i28 .T?810
38
23
.r^fiH .'^]."^50
.59318
.80607
.60n4
.79459
.6
*1 .77802
87
JM
.r.T:ris .-j.m:}
.69342
.80489
.60788
.79441
.6
173 .77278
38
2E
.r.7l'.V,> .KML^S
.59365
.80472
.60761
.79424
.6
196 .77255
35
2::
.r.:!«:'; .sisr')
.59889
.80455
.607«4
.79406
.6
>18
.T?286
34
27
.n?.fiftJ .K!lr,^:J
.69112
.80438
.60807
.79888
.6
>40
.77218
88
23
.fir^riii .s[((,5
.60436
.80420
.60830
.79871
.6
.63
.7n99
88
29
.f.xij? •'':■.■■}
.69459
.80403
60853
.79363
1.6
85
.rn8i
31
30
.; J
.59482
.80386
.60876
.79885
.6
t08
.77162
80
81
.1 >
.69606
.80368'
.60699
.79818
'.6
190
.77144
20
82
.1 }
.69529 .80351
.60922
.79300
.6
153
.77125
28
88
.( ij
.69552 .80334;
.60945
.79282
.6
(76
.77107
27
84
.( i'
.59570 .80316;
.60968
.79264
.6
!98 ;. 77088 i 28
86
J r
.69599 .80299
.60991
.79247
.fl .-: '
' . ■
.20
.77070, 25
86
J )
.59622 .80282
.61015
.7TO29
.6-:.->
> 1 ."
i.:;;'42
.77061 1 24
87
.1 \
.596461.80264
.61038
.79211
.6:3 1 1
. .".i..i
.,.;,'65
.77083 23
88
.[^■'■'^N ,MJ.,J
.59669 .80247
.61061
.79193
.e-r::-;
/;><i[<t
,t^i:87
.77014 22
89
.fpviix'f h]-iv)
.59698 .802301
.61084
.79176
.e;):.u
.7Ht:jh
Ji'iSlO
.76996 21
40
.fK^Lor >i-)-j
.59716
.802121
.61107
.79168
.tiiW^
.Tf^n^j
.t5;iS82
.76977|20
41
J J
.59789
.801951
.61130
.79140
.6
^64
.78959 19
42
A J
.59763
.80178
.61153
.79122
.e
177
.76940 18
48
.[ I
.69786
.80160
.61176
.79105
.e
{99
.76081 17
44
.[ 1
.59609
.80143
.61199
.79087
.6
62
.76903 18
45
.i r
.59832
.80125!
.61222
.79069
.€
144
.76884116
46
.(/. y
.59&'')6
.801081
.61245
.79051
.c
>66
.76866114
47
.hM\-^ .^11?.}
.59879 .800911
.61268
.79033
.fl
169
.76847' 13
48
.r^-^j'.j^i M]iw5'
.59902
.800781
.61291
.79016 ' .e
111
.76828 12
49
.r.>-..M'.i ^^.'.J
.59926
.80056
.61314;. 78996 .fl
1.B3 .76810 11
50
.1 8
.59949
.80038
.61337
.78960
.e^raK.
:i7W,
.WLI66 .76791 10
U
.68567
.81055
.69972
.80021 '
.61360
.78962
.tr.s^
,77?*70
.6-1(^78 1.76772 9
52;. 58590
.81088
1.59995
.80003
.61383
.78944
.6; -1
.77>*1
.ft) iOO .76754! 8
58 .58614
.81021
1.60019
.79986
.61406
.78926
.62774
.rrma
.&I123I. 76785; 7
54 .58687
.81004 1 .60(V12
.79968
.61429
.78908
.6':'.'n^^
.nmi
.fi-1 145 1.76717 6
55 .58661
.80987 ' .60065
.79951
.61451
.7S891
.6'/Ni9
.77RO0
.©11671.76698 6
56 .58684
.80970 .60089
.79934
.61474
.78873
.6->ie
.77788
.*i* 190 .766791 4
57 , .58708
.80953 .60112
.79916
.614971.788551
.6>iit
.TTTflQ
.6^l^'12 .76661 1 8
58
.58781
.80936 {.60136
.79899
.61520
.78837
.6-,'^-^7
.777M,
Jl4iJ34 .766421 2
59
.68766
.80919
.60158
.79881
.61543
.78819 l.6-'.K>
TTTBa:
mm .78628
1
60
.68779
.80902
Sine
.60182
Cosin
.79864
Sine
.61566
.78801
Sine
tCostn
1 61
.77715' ,^IJ79 .76604
sm|ia^''Sine
0
■^1
Cosin
e
M- 1
63» 1
62« 1
L* 1
1 6<
Y
788
SURVEYING,
TABLE V\,— Continued.
Natural Sines and Cosines.
43«
Cosin
Sine
.74314
.68300
.74295
.68221
.74276
.68212
.74256
.68204
.74237
.68285
.74217
.68306
.74198
.68327
.74178
.68349
.74159
.68370
.74139
.68391
.74130
.68412
.74100
.68484
.74080
.68455
.74061
.6&476
.74W1
.68497
.74023
.68518
.74002
68580
.73983
.68561
.7890.3
.68588
.78944
.68608
.739SU
.686S4
.73904
.68645
.73885
.68006
.73865
.68088
.73846
.68709
.78820
.68780
.73806
.68751
.73787
.68772
.73767
.68793
.73747
.63814
.78788
.68885
.7aw>8
.68857
.73688
.68878
.73869
.68899
.73649
.68930
.73029
.68941
.73610
.68902
.73500
.68983
78570
.69004
.73551
.69025
.78531
.69046
.78611
.69067
.73491
.69088
.73473
.69109
.73152
.69130
.73432
.69151
.73413
.69172
.73893
.69193
.78373
.69214
•Taa-vji
.69235
.78333
.09256
.73314 1
.69277
•2?nli
.69298
Cosin
32'
7:^254
.7321
.7:il»'>
.7317.-1
. 73155
Sine
.72957
.72987
.72917
.72807
.72877
.72857
.72817
,72797
.72777
.72757
.72787
.72717
.73097
.72677
.72657
.72037
.73017
.72597
.72577 1
.72557
.72587
.72517
72497
.72477
.72457!
■2437
.72417
73397
.72377
.72357
72887
440
.60904
.69925
.69946
.69966
.60967
.70006
.7t)029
.710049
.70070
.700U
.70857
.70277
.TONS
89
.69319
.093401
732:M .693fil,
.693H2
.09KW
.69424
.69445
.O&iOOi
Cosin
.70619
.70889
,70860
,70881
.70401
722HJ^I.70422
,7219<>,. 70448
,72176 .70468
72150 1.70484
,72130 .70506
,721 16' 1.70625 1.'
.72095 .TOMO .'
.73075 .70567 ."
,73055 .70587|.'
.72085 .70608 .'
.72015 .70838 .'
.7l91i-» 1.70649 .'
.71'j74 1 1.70670 .
. 71 954 1 1. 70890 .
.71934 .70711 J
Sine ' Cosin 1 1
71100 19
.71090 18
.71060 17
.71069 16
.71010^ 15
70999 14
7ro78,'l8
.70957 12
iBTl 11
.70916 10
TABLES,
789
TABLE VII.
Natural Tangents and Cotangents.
O* 1
!• 1
2<» 1
30 1
/
0
Tang
.00000
Cotang
Tang
.01746
CotangI
Tang
.03492"
Cotang !! Tang
Ck)tang
In Unite.
57.2900
28.6363
.05)^1
19.0811
60
1
.00039
8437.75
.01776
56.8506
.03521
28.8994
.05270
18.9755 l69
2
.00058
1718.87
.01804
56.4416
.03550
28.1664
.05299
18.8711 168
8
.00087
1145.92
.01888
64.6618
.08679
27.9378
.05328
18.7678 67
4
.00116
a59.436
.01862
68.7086
.03609
27.7117
.05357
18.6666 |66
6
.00145
687.549
.01891
62.8821
.08688
27.4899
.05387
18.5645 65
6
.00175
572.957
.01920
62.0807
.03667
27.2715
.0W16
18.4645 64
7
.00204
491.106
.01949
61.3032
.03696
87.0566
.06446
18.8656 158
8
.00233
429.718
.019ffi
60.M86
.03^5
86.^50
.05474
18.2677 162
9
.00263
881.971
.02007
49.8167
.03754
26.6867
.05506
18.1706 61
10 .002dl
848.774
.02066
49.1068
.08788
28.4316
.06688
18.0750
50
11 ! .OOSSO
812.621
.02066
48.4181
.08812
86.2296
.06668
17.9608
49
12 .0(»4»
286.478
.02096
47.7886
.06842
26.0607
.05591
17.8868
48
13 .00378
264.441
.02124
47.0668
.06871
86.8348
.06620
17.7984 '47
14
.00107
245.552
.02153
46.4489
.08900
25.6418
.05649
17.7015 46
15
.00136
229.182
.03182
45.8284
.03928
25.4517
.06678
17.6106 45
10
.00466
214.858
.02211
45.2261
.08958
25.2644
.06708
17.6206 44
17
.00495
202.219
.02240
.02269
44.6386
.08987
26.0796
.057^
17.4814 48
18
.00524
190.984
44.0661
.04016
84.8978
.05766
17.8482 42
19
.00553
180.932
.02298
.0282§
43.5081
.04046
84.7185
.06796
17.2568 41
20
.00682
171.885
42.9641
.04075
84.6418
.05824
17.1606 40
2t
.00611
168.700
.02857
42.4885
.04104
84.3675
.05854
17.0687 39
22
.OO&IO
156.259
.02386
41.9158
.04188
84.1957
.05888
16.9990 ;38
23
.00669
149.465
.02415
41.4100
.04162
84.0268
.06018
16.9150
87
21
.00696
148.287
.02444
40.9174
.04191
88.8598
.06941
16.8810
86
25
.00727
.00756
137.507
.02473
40.4358
.04220
83.6945
.06970
16.7496
86
26
132.219
.02508
89.9665
.04250
88.5321
.05999
16.6661
84
27
.00785
127.821
.02531
89.5059
.a4279
23.8n8
.06029
16.5874
88
28
.00815
122.774
.02560
89.0568
.04808
28.2137
.06058
16.5075
82
29
.00844
118.540
.02589
88.6177
.04387
83.0577
.06087
16.4288
81
30
.00878
114.689
.02619
38.1886
.04866
82.9068
.06116
16.3499
80
81
.00902
110.892
.02648
87.7886
.01395
22.7519
.06146
16.2722
29
32
.00931
107.426
.02677
87.8579
.04424
22.6020
.06175
16.1952 28
33
.00960
104.171
.02706
88.9560
.04454
28.4541
.06204
16.1190 27
31
.00969
101.107
.02735
88.5627
.01483
22.3061
.06238
16.0436
26
35
.01018
98.2179
.02764
86.1778
.04512
22.1640
.06262
16.9687
26
36
,01047
95.4895
.02798
85.8006
.04541
82.0217
.06291
15.8946
84
37
.01076
92.9086
.052822
85.4313
.04670
21.8813
.06821
15.8211
28
88
.01106
90.4688
.02861
85,0695
.04599
21.7426
.06850
15.7488
22
89
.01186
8«.1436
.02881
84.7151
.04628
21.6056
.06379
16.6762 21
40
.01164
85.9398
.02910
84.3673
.04658
21.4704
.06406
16.6048 20
41
.01193
88.8485
.02939
84.0273
.04687
81.8869
.06487
16.5840 19
42
.01222 1 81.8470 1
.02968
88.6985
.04716
21.2019
.06467
16.4688 18
43
.01251
79.M34
.02997
88.8662
.04745
21.0747
.00496
15.8948 17
44
.01280
78.1268
.03026
88.(M52
.04774
20.9460
.06525
15.8264 16
45
.01909
76.8900
.03055
82.7308
.04803.
20.8188
.06554
15.2571 15
46
.C1338
74.7292
.03064
82.4218
.04833
20.6983
.06584
15.1898 14
47
.01367
73.1890
.03114
82.1181
.04862
20.5091
.06613
15.1822 13
48
.01396
71.6151
.03143
31,8206
.04891
20.4465
.06W2
16.0667
12
49
.01426
70.1533
.03172
8l.52»4
.04920
20. .3258
.06671
14.9898
11
50
.01455
68.7501
.08201
81.2416
U)49i9
80.2056
.06'i'00
14.9244
10
51
.01484
67.4019
.08280
80.9599
.01978
20.0878
.08780
14.8596
9
52
.01513
66.1056
.08269
80.6883
.05007
19.9702
.06759
14.7964
8
53
.01542
6t.S580
.08288
80.4116
.05087
19.a546
.06788
14.7817
7
M
.01571
63.6567
.(»317
80.1446
.05066
19.740:J
.06817
14.6686
6
55
.01600
62.4992
.08346
29.8838 1
.05095
19.6273
.06*47
14.6059
6
56
.01629
61.3829
.08376
29.6245 1
.05124
19.5156
.06876
14.5488
4
57
.01658
60.3068 '
.08406
29.3711 i
.05153
19,4051
19.d959
.06905
14.4828
8
68' .01687
59.2659
.03434
29.1220
.05182
•oea^
14.4212
2
5'j .oina
68.2612
.0846:}
28.8771
.06212
19.1879
; .06963
14.3607
1
00
.01746
Cotang
67.2900
.03492
Cotang
28.6368
Tang
.05241
CcUng
19.0811
1 .06993 14. .3007
0
I'ang
Tang
, CotangI Tang
89»
1 88<»
. ■ 87* d|
gitized g
6° 1
790
SURVEYING.
TABLE Vll.-^CtmHMued,
Natural Tangents and Cotangents.
i
4«
5« 1
6« 1
7-
n
Tang
Cotanj?
Tang
.06749
Ootang
Tan^
Cotang
Tang
Ootang
'
.06998
14.8007
11.4801
.10510
9.61486
".12878
6.14486
w
1
.07082
14.2411
.08778
11.8019
.10640
9.48781
.12806
8.18481
09
2
.07051
14.1821
.06807
11.8540
.10660
9.46141
.12888
8.10686
68
8
.07080
14.1286
.08887
11.8168
.10609
9.48516
.12867
8.06000
67
4
.07110
14.0655
.06866
11.2780
.10628
9.40004
.12897
8.06674
66
5
.07189
14.0079
.08896
11.2417
.10667
9.88807
.12426
8.04756
66
6
.07168
18.9607
.06925
11.2048
.10687
9.85724
.12466
8.08848
64
7
.On97
18.8940
.06964
11.1681
.10716
9.88156
.12485
8.00948
68
8
.07227
18.8378
.06968
11.1816
.10746
9.80509
.12616
7.99068
68
9
.07856
18.7821
.09018
11.0954
.10775
9.28058
.12544
7.97176
51
10
.07286
18.7267
.09042
11.0694
.10605
9.26680
.12674
7.96808
60
n
.07814
18.6n9
.09071
11.0287
.10684
9.28016
.12606
7.98486
49
u
.07844
18.6174
.09101
10.9682
.10668
9.20516
.12688
7.91582
48
18
.07878
18.5684
.09180
10.9629
.10608
9.18028
.12662
7.80784
47
14
.07402
18.5096
.00150
10.0178
.10922
9.15554
.12602
7.87886
46
16
.07481
18.4566
.09180
10.8829
.10962
9.18098
.12782
7.86064
46
16
.07461
18.4089
.09218
10.8488
.10961
9.10646
.18751
7.84848
44
17
.07490
18.8516
.09247
10.8189
.11011
9.08211
.12781
7.aM28
48
18
.07519
18.2996
.09277
10.7797
.11040
9.05789
.12810
7.80«8
48
19
.07548
18.2480
.00806
10.7457
.11070
9.06879
.12840
7.78886
41
90
.07678
18.1960
.09885
10.7119
.11090
9.00988
.12809
7.77066
40
«
.07607
18.1461
00865
10.6788
.11128
8.98696
.12809
7.78864
89
22
.07686
18.0958
;09894
10.6450
.11158
8.96227
.12989
7.78480
88
28
.07666
13.0456
.00428
10.6118
.11187
8.98867
.12956
7.71716
87
24
.07606
12.9962
.09458
10.5780
.11217
8.91520
.12968
7.60067
86
25
.07724
12.9469
.09488
10.6462
.11246
8.89185
.18017
7.68806
86
26
.07758
12.8981
.09511
10.6186
.11276
8.86862
.18047
7.66466
84
27
.07782
12.6496
.09641
10.4818
.11806
8.84561
.18076
7.64788
8S
28
.07812
12.8014
.09670
10.4491
.11886
8.82262
.18106
7.68006
88
20
.07841
13.7536
.09600
10.4172
.11864
8.79964
.18186
7.61887
81
80
.07870
12.7062
.09629
10.8854
.11894
8.77689
.18166
7.60676
80
81
.07809
12.6591
09658
10.8688
.11428
8.76426
.18195
7.67878
89
82
.07929
12.0124
.09668
10.8224
.11462
8.78172
.188M
7.66176
88
88
.07958
12.6660
.00717
10.2918
.11482
8.70081
.18854
7.64487
87
84
.07987
12.6190
.t)9746
10.2602
.11611
8.68701
.18284
7.68806
86
85
.08017
12.4742
.09776
10.2294
.11541
8.66482
.18818
7.61188
85
86
.08046
12.4288
.09606
10.1968
.11570
8.64275
.18848
7.49466
84
87
.06075
12.8888
.09884
10.1688
.11600
8.62078
.18838
7.47806
28
88
.08104
12.8390
.09864
10.1881
.11629
8.59698
.18408
7.46154
88
89
.08184
12.2946
.09698
10.1080
.11660
8.67718
.18488
7.44509
81
40
.08168
12.2506
.09028
10.0780
.11686
8.66665
.18«n
7.48871
80
41
.0R192
12.2007
.00062
10.0188
.11718
f 58402
.18481
7.41840
19
42
.08221
12.1682
.09981
10.0187
.11747
8.51860
.18681
7.89616
18
48
.08861
12.1201
.10011
9.98081
.11777
8.49128
.18660
7.87900
17
44
.06280
12.0772
.10040
9.96007
.11806
8.47007
.18680
7.86889
16
46
4)8809
12.0846
.10069
9.98101
.11886
8.44896
.18609
7.84786
16
46
J^»
11.9923
.10099
9.90211
.11866
8.42796
.18689
7.88190
47
.06868
11.9504
.10128
0.87888
.11896
8.40705
.18609
7.81000
48
.08897
11.9087
.10158
9.64482
.11924
8.88625
.18698
7.80018
49
.06427
11.8673
.10187
9.81641
.11954
8.86655
i .18788
7.88448
60
.06450
11.8262
.10216
9.78817
.11968
8.84496
.18768
7.86878
51
.06485
11.7868
.10246
8.76009
.12018
8.82446
.18787
7.86810
52
.08514
11.7448
.10276
9.78217
.12042
8.80406
.18817
7.88754
58
.06544
11.7M5
.10806
9.70441
.12072
6.28876
.18846
7.28804
U
.(W578
11.6646
.10684
9.67680
.12101
8.26856
.18876
7.80061
551 .(fe602
11.6248
.10068
9.64985
.12181
8.24846
.18906
7.19186
56 j 08682
11.5858
.10898
9.62206
.12160
8.22844
.18986
7.17504
57* .08661
11.5461
.10422
9.59490
.12190
8.20852
.18966
7,160n
58, .0H690
11 5072
.10452
9.56791
.12219
8.18870
.18996
7.14568
50 .08720
11.4685
.10481
9.54106
.12249
8.16396
.14084
7.18048
60
/
.08749
Ootang
11 4801
.10510
Ootang
9.51486
.12278
8.14485
.14064
7.11587
Tang
Tang
Cotang
Tang
Ootang
Tang
7
h^
1 8
5»
a
4*
8
30
a
»•
le
TABLES.
791
TABLE V\\,^Continued.
Natural Tangents and Cotangents.
' 4
8»
r e*
10»
It 11-
60
,1«)M
Cotanir
Twig
,16888"
Cotftog
.17633 IT 1171 JH"
Taug
ICotang
7.11537
O.S1875
ll^tSS
5.14455
1
.14084
7.10088
.15868
8.9oise
.17065
B.fl*J165
.19488
5.13658
50
%
.HI 19
7.08M9
.15888
a.»or^
.17B03
6,toiJ05
.19498
5.12869
58
s
.14149
7.(m»e
1 .16088
6.27880
.ITTiSa
.1I1699
6.12000
57
4
.14173
7.«a79
.]69fi8
6.«60&6
.17758
5,ftft»96
.196SO
6.11979
56
5
.i*»e
7{M105
.I508S
0.26480
.17785
5 63344
.19560
5.1M90
55
B
J43^
7.03837
.16017
O.lMSei
.17813
6.61397
.10619
6.00m
54
7
AASm
7.01174
.16017
6.23160
.17SI9
5.6045?
.19619
5.080S1
5.3
8
.14a»l
fl. 99718
.10077
6.)S8003
,17878
6.59&U
.19680
5 081.T0 ,;53|
9
,14ftil
6-9^6&
.16107
0.90651
.17908
5.S6679
.19710
S 07360
51
10
J4S&I
6.90ai8
.16187
0.19708
.17988
5.67638
.1!}740
5.00584
.W
11
.143S1
«,Q538{1
.10107
C.18&50
.17GS3
n.fiOTOfl
.J9770
sa**,^
49
la
14410
6.B39B8
.IfllOfl
ft 174T9
J7903
5.65777
,19801
5.0GO37
.18
la
,14440
e.aasss
je^
6 lfiiJ83
.18023
5.54851
.10881
5 04307
47
M
44I7I>
e.9ll04
.16256
€.15151
.18058
5.58927
.19861
5.03499
46
15
J4I00
0.86088
.10380
6J403S
.18088
5.53007
.19801
6.0tf734
45
le
.1*538
6.8a279
.16S10
6.18899
.18113
6.5t3090
,10601
6.01971
41
17
.145&0
6.86874
.10»I6
0.11779
.18148
5.51176
.19992
6.01210
43
IS
.14568
6.85475
.16570
0.10661
.18173
5.a^)an
.10083
6.00151
m
lftl
.14418
6.&i0®
J04^
6.095GfiJ
.I830a
5 4(ia56
.20012
4.09696
41
m
.14S4S
^mm
.10435
6.08444
.18233
5.48451
.20013
4.98040
40
21
.14878
fl.8i9ia
,1M85
OOTTMO
JR263
6.47Me
.30073
4.98188
90
2^
.14707
6-79996
.164lfl
fi.lNj'JlO
.I8in«
6,40r48
,20103
4 i>7438
3^^
29
,14717
e.ra564
.165ii5
0.05143 .
.1S323
6.45751
,2l)l$J
4.96690
37
24
.147i7
6.77199
.1«5S5
6 044)51
.18353
B.44857
.30101
4.95945
86
26
.HTM
fl.75838
.16585 1
0.02S62
.1^84
8.48966
,90194 ,
4.06^
35
40
J4a«
6.744M
.10015
6.01878
.18114
6,48077
.SftiJi
4.04460
%
27
.148M
«.rai33
.10046
6.00797
,18444
B.43193
.a<>J54
4 93?>1
^1
J48W
6.71799
.16674
5.00790
.18474
5.41309
.2thiB5
A y VTXS4
ai;
»
.11915
fl.TWflO
,wm
6.ge«Mo
JBS04
6.40429
' .2(015
A\tii\%
31
m
J4M&
6.M116
.167ai
5.97678
.18684
6.«B6ei
.30315
4.91516
30
51
.14975
6.67787
,16764
5 D66I0
.18BS4
6.98877
.S0970
4.90785
ffl?
Si
,15006
t.mm
.vsm
5.95418
,18604
5.971W
.S04O6
4.90056
M
38
.15094
0.66144
.16884
6.94300
.18624
8.88996
.20436
4.80680
2T
94
,lfiOSl
0.63831
,16864
6.08986
.18654,
5,30070
.2CU66
4,88606
96
S5
.160B4
G.eSSS9
.16881
6:ee283
.1S684
6.35008
.201^7
4.87882
26
98
.lfil»4
O.fllfig
.16«14
5.01236
.18714 '
6.34*46
.mi^
4.87169
M
J7
.15168
6.5aosn
.16044
5.90191
.18745
5.9S1S7
.a(Kfj7
4.80444
28
88
.15183 ,
6.6aaa7
.16974
6.89151
.38775
5.S3631
.20588
4.857^7
92
9ft
.15219
6.57339
.17004
6.881t4
.18805
5.317^^
,20$] 8
4.83013
21
m
.15»43
0,50066
,17083
6.87080
.1683S
B. 30928
.a0648
4.84300
20
41
laara
6.54777
.17063
B.880B1
.18868
6,30080
,90070
4.Rafi90
19
43
.IKKiS
6.5S60a
,17093
s.asossi
.18888
5.29895
.80700 !
4.8!J883
18
4a
.15833
6.SSB4
.171!83
B.810O1
.18925
6,28393
,90730
4.8:^175
17
44
.15303
6.5097O
.1716.1
5 820R3
.18955
5.37553
,90770
4.8N71
16
is
15391
6.49710
,17183
S.fllDtW
.18986
5.36716
.90800
4.80769
15
46
15431
6.4Si6d
,17»ia 1
5.80958
.19010
6.25880
.20890
4.80068
14
47
15451
0.47806
.17*48
6.7BM4
.19046
6.25018
.00861
4.79870 ,
19
4S
.15481
0.45001
J7S7a
6,78088
.19076
0.24318
.80801
4.78678
12
40
.15611
0.44790
.17908
6,7799(1
.19106
5.33391
.20021
t1
50
.15540
0.4&I84
.ima
6.78087
.19136
5.1^566 1
.90052
4;773afl
10
51
.15570
6-4S3i»
J7368
6.7M41
.19186
5 21744
.20982
4.78506
0
M
.158fW
e.41CK6
.17398
6.74940
.m»7
5 20025
.21013
4 75906
8
59
,15630
0.a«8(M
.17428
6.73980
.19237
530107
.2lf^3
4.75219
7
&i
.15M0
0.3Si87
.17453
B.7S974
.192S7
5.1i*V9?J
.21073
4 74531
0
OS
.15B8B
0.S7374
.17483
6.71992
.19387
5.IS4«)
.21 IW
4.73861
6
u
.ISTlft
036105
.17518
6.71013
.19317
5,17671
.811*1
4.78170
4
w
.15749
6.94961
.17513
67twr
J9^7
5.168S8
.21164
4.7»I90
8
w
.15779
0.33761
.17573 1
6090&1
.19878
6.16068
.21195
4.71813
9
B»l
.1580*
03»«6
.17808
6 88094
.1OI08
6.15e5«
.21935
4.71187
1
00
J58»
691333
.17883
CotAQg
6.071)88 ,
.19498
6.144&0
.21256
4.70469
_0
OoUqc
Taog
T&dg
CotAQf
l'*iig
Tang
il- 1
80- 1
79- 1
7a. 1
■gitized by
Goojle
^92
SURVEYING.
TABLE VW.—Continued,
Natural Tangents and Cotangents.
1
20
Tang
Cotong.
0
.21256
4.70468
1
21286
4.69791
£
21816
4.69121
8
21817
4.68452
4
21877
4.67786
6
■21408
4.67121
«
.21438
4.66458
7
21409
4.65797
8
.21499
4.65138
y
.216^9
4.64480
10
.21560
4.68825
n
.21590
4.68171
12
.21621
4.62518
18
.21651
4.61868
M
.21682
4.61219
15
.21712
4.60572
16
.21748
4.69927
17
.21778
4. .59283
18
.21804
4.58641
19
.21834
4.58001
«)
.21864
4.57368
21
.21895
4.56726
9&
.21925
4.56091
28
.21956
4.5M58
£4
.21986
4.54826
25
.22017
4.54106
26
.22M7
4.535G8
27
.22078
4.52941
28
.22108
4.52316
29
.22130
4.51698
80
.22160
4.61071
81
.22200
4.50451
82
.22281
4.49882
83
.22261
4.49215
84
.22292
4.48600
85
.23822
4.47986
86
.22358
4.47374
87
.22383
4.46764
88
.22414
4.46155
89
.22444
4.45548
40
.22475
4.44942
41
.22605
4.44888
42
.22536
4.48785
43
.22567
4.48134
44
.22597
4.42534
45
.22628
4.41986
46
.22658
4.41340
47
.22689
4.40745
48
.22719
4.40152
49 .23750
4.89560
60
.22781
4.88009
61
.22811
4.88881
53
.22842
4.87798
53 .22872
4.87207
M .22908
4.80623
65 .22934
4.36010
661 .239&4
4.85459
57
.22995
4.34879
58
.23026
4.34300
69
.23056
4.83723
60
.23087
Cot&ng
4.83148
/
Tang
7
7«
18« I
Tang
.23067
Cotang
4.88148
.33117
4.82678
.23148
4.82001
.23179
4.81430
.28200
4.80860
.28240
4.80291
.28271
4.29724
.28801
4.29169
.23888
4.28605
.23368
4.28032
.28898
4.27471
.28424
4.20911
.28456
4.26852
.23485
4.26795
.23516
4.25289
.23547
4.24685
.23578
4.24132
.28608
4.23580
.23639
4.28030
.28670
4.22481
.28700
4.21988
.28781
4.21887
.28768
4.20842
4.20298
.28823
4.19756
.28854
4.19215
.23885
4.18675
.23916
4.18137
.23946
4.17600
.23977
4.17064
.24008
4.16680
.24039
4.15997
.24069
4.15465
.24100
4.14984
.24131
4.14405
.24162
4.13877
.24198
4.13350
.24233
4.13825
.24254
4.13301
.24285
4.11778
.24316
4.11256
.94847
4.10786
.24877
4.10216
.24408
4.09699
.24439
4.09182
.24470
4.08666
.24501
4.08162
.24532
4.07639
.24562
4.07127
.24598
4.06G16
.24624
4.06107
.24655
4.06599
.24686
4.05092
.24717
4.04586
.24747
4.04081
; .21778
4.08578
.24809
4.03076
j .24^40
4.02.574
.24871
4.02074
1 .21909
4.01576
.21933
401078
Cotang
Tang
7
^
14»
Tang
.24933
.24904
.24996
.25026
.26066
.25067
.26118
.26149
.26180
.25211
.26242
.25278
.25304
.26335
.25397
.25428
.26459
.25490
.26521
.26662
.25614
.25645
.25678
.28707
.26788
.25769
.25800
.25831
Cotang
.25955
.26017
-26048
26079
.26110
.28141
.26172
.26297
.26800
.26421
.26453
.2^88
.26515
.26546
.26577
.26639
.26670
.36701
.26783
.26':-64
.26795
Cotang
4.01078
4.00682
4.00086
3.99502
8.99099
8.96607
8.98117
8.97627
8.9n89
8.96661
8.96166
8.96680
8.95196
8.91713
8.94232
8.98751
8.93271
8.92798
8.92316
8.91839
8.91864
8.90890
8.90417
8 89945
8.89474
8.89004
8.88536
8.88068
8.87601
3.8n36
8.86671
8.86206
8.85745
8.85284
3.84834
8.84304
8.83906
8.88449
8.82992
8.88537
8.83088
8.81680
8.81177
8.80ra6
8.80276
8.79857
8.79378
8.78931
8.78485
8.78040
8.77505
8.77153
8.76709
8.76268
8.76838
S.75388
S. 74950
8.74512
8.74075
8.78640
3.78205
Tang
W
^uig
.26795
.26030
.26051
.27018
.270^4
J87076
.27107
.27188
.27169
.27801
.27832
.27268
.2^3294
.2';826
.27857
.27888
.37419
.27451
.37483
.27518
.37545
.27576
.27007
.27688
.27670
.37701
.37783
.27764
.27795
.37826
.27858
.27889
.27931
.37953
.37968
.28015
.28046
.28077
.28109
.28140
.28172
.28303
i Cotang
8.78n5
8.73771
8.73888
8.71907
8.71478
; 8.71016
3.70616
8.70186
8.007)61
8.
8.
.28297
.28300
.28891
.28428
.28454
.28480
.28517
.28549
.28612
.28648
.28675
Cotang
8.68485
8.66061
8.67688
8.67217
8.66796
8.66S76
8.65057
8.66588
8.65131
8.64706
8.64280
8.68874
8.68461
8.63048
8.68686
8.63894
8.61814
8.61405
3.60096
860668
8.60181
8.697TO
8.59870
3.58066
8.68602
8.68160
8.67758
8.57867
8.66957
8.56557
8.68160
8.66761
8.56864
3.54968
8.51678
8.54179
8.5371B5
8.58896
8.53001
8.68600
8.58319
3.51829
8.61441
8.51058
8.50666
8.60279
8.49604
8.49609
8.49125
8.48741
Tang
75«
74*
trnogle
Digitized by
TABLES,
793
TABLE VW.^-QmHnued,
Natural Tangents and Cotangents.
ie»
.28864
.28927
.28958
.29274
.29400
.2M63
.29495
0 .28675
1 .28706
2 .287:«
3 .28769
4
5
6
7
8
9
10
11 .29021
12 .29053
18! .29084
14 .29116
15 .29147
16 .29179
17 .29210
18
19
20
21
.29590
.29621
.29658
.29716
.29748
.29780
.2981t
.29848
.29875
.29988
.29970
.30001
.80083
.80065
.80097
.80128
.80160
.80192
.80255
.80287
.80319
.80351
.30414
.80446
.80478
.80609
.80641
.30673
Cotang
I Cotang '
3.48741 i
[8.48360 I
8.47977
8.47596
8.47216
8.46837
8.16458
8.46060
8.45703
8.46327
3.44051
8.44576
3.44202
8.48829
8.43456
3.48084
3.42713
8.42813
8.41978
3.41604
8.41236
8.40869
8.40002
8.40186
3.89771
8.39106
8.89042
8.88679
8.88817
8.87956
8.87594
8.37234
8.36875
8.36516
8.36158
3.85800
3.35148
8.86067
8.34782
8.8137r
8.34023
8.83670
8.a3317
8.82966
8.32614
8.32264
3.81914
8.81565
3.81216
8.30868
8.30621
8.80174
8.29829
3.29488
8.29189
3.28795
8.28452
8.28109
8.27767
8.27426
8.27086
Tang
78»
W
Tang
Cotang
.80573
8.27085
.80605
8.2ff745
.30637
8.26406
.30669
8.26067
.80700
8.25729
.80732
8.26892
.30764
8.25055
.30796
8.24719
.30828
8.24888
.80860
8.24049
.30891
8.23714
.80928
8.23381
.80955
8.23048
.30987
3.22715
.81019
8.22384
.31051
8.22058
.31068
8.21722
.81115
8.21392
.81147
8.21063
.81178
3.20784
.81210
8.20406
.81242
8.20079
.81274
8.19752
.31806
8.19426
.81338
8.1U100
.31370
8.18775
.81402
3.18451
.81434
8.18127
.31466
8.17804
.81498
8.17481
.81530
8.17159
.81562
8.16838
.31594
8.16517
.81626
8.16197
.31658
8.15877
.31690
3.15558
.31722
3.15240
.81754
3.14922
.81786
3.14605
.31818
8.14288
.81860
8.13972
.81882
3.13656
.81914
3.13341
.31946
8.13027
.31978
3.12713
.32010
8.12400
.32042
3.12087
.32074
3.11775
.32106
8.11464
.32139
8.11153
.82171
8.10842
.32203
8.10582
.32285
3.10228
.82267
3.09914
.82299
8.09606
.82331
8.09298
.82368
8.06991
.82396
808685
.82428
8.06879
.32460
8.08073
.32492
8.07768
Cotang
Tang
18»
Tang
.82556
.32621
.82653
.32717
.82749
.82814
.82&46
.82911
.32975
.33007
.33040
.33072
.33104
.33136
.83169
.88201
.83266
.83298
.33363
.33395
.3*427
.83460
.83492
.835£^4
.83557
.83589
.83654
.83718
.38751
.88788
.83816
.83&18
.33881
.33913
.33945
.88978
.84010
.84043
.34075
.84108
.34140
.34173
.84205
.34238
.84270
.84303
.84335
.84308
.84400
.34433
Cotang
Cotang
3.07768
3.07464
8.07160
8.06857
8.06554
8.06252
3.05960
8.06649
8.05849
8.06049
8.04749
8.04450
8.04152
8.03854
8.03556
8.03260
8.02968
8.02667
8.02372
8.02077
3.01783
3.01489
3.01196 I
8.00903
3.00611
3.00819
19«
2.99788
2.99447
2.99158
2.98868
2.98580
2.98292
2.98004
2.97717
2.97430
2.97144
2.96868
2.C6573
2.96288
2.96004
2.95721
2.95487
2.95155
2.948?^
2.94591
2.94809
2.94028
2.93748
2.93468
2.98189
2.92910
2.92632
2.92354
2.92076
2.91799
2.911^6
2.90971
2.90»)96
2.90421
Tang
72*
71»
Tang_
.34438
.84465
.84498
.84530
.34563
.84596
.84661
.34726
.34758
.34791
.84^4
.84856
.84889
.84954
.84987
.86020
.85066
.86118
.86150
.86188
.85216
.86281
.86814
.86846
.86879
.85412
.85445
.85477
.85610
.35543
.35576
.85608
.85641
.85674
.337t)7
.85740
.85772
.35805
.35871
.35904
.35987
.86002
.86035
.C0068
.36101
.36134
.36167
.86199
.86232
.36265
.86831
.86364
.36397
Cotang
Cotang
2.90421
2.90147
2.89873
2.89327
2.89055
2.88783
2.88511
2.88240
2.87970
2 87700
2.87430
2.87161
2.86892
2.86624
2.86.356
2.8G089
2.85822
2.85655
2.89289
2.86023
3.84758
2.84494
2.&1229
2.a396f)
2.8:^702
2.83439
2.83176
2.82914
2.82391
2.82130
2.81870
2.81610
2.81350
2.81091
2.80574
2.80316
2.80059
2.79802
2.79546
2.79289
2.79033
2.7877«
2.7^523
2.78269
2.78014
2.77761
2.77507
2.77^254
2.77002
2.76750
2.76198
2.76247
2.75996
2.75746
2.75496
2.75246
2.74997
2.74748
Tang
TQoy i^cbq^Ie
794
SURVEYING.
TABLE VW.—ConHnutd.
Natural Tangents and Cotangents.
1
"0
w \
«!• I
M- 1
W
60
Tang
Cotant
Tang
Ootang
Tan^
Ck>tang
Tang
.86897
8.74748
.38386
8.60609
.40408
8.47509
.48447
1
.86480
8.74499
.88480
8.60888
.40486
8.47808
.48468
8. 5
60
8
.86468
8.74861
.38458
8.60067
.40470
8.47095
.48516
8. a
58
8
.86496
8.74004
.38487
8.50681
.40604
8.46886
.48651
8. 1
57
4
.86589
8.78756
.38520
8.58606
.40538
8.46668
.48585
8. r %
56
6
.86568
8.73609
.88558
8.50881
.40678
8.46476
.48619
8. i' 4
66
6
.86506
8.78268
.38587
8.50156
.40606
8.46870
.48654
%:nm
54
7
.86628
8.78017
.38620
8.58038
.40640
8.46065
.48666
8.:: 1-4*8
58
8
.86661
8.78771
.38654
8.58708
.40674
8.45860
.48788
8.:uiw9
58
9
.86694
8.72526
.38687
8.58484
.40707
8.45665
.48757
8.:';h-^;1
51
10
.86787
8.72281
.38721
8.58861
.40741
8.45451
.48791
%.^^^%\
50
11
.86760
8.78086
.88754
8.56088
.40275
8.4SM6
.4an6
8.88806
40
18
.86798
8.71798
.38787
8.57815
.40609
8.46048
.48660
8.88817
48
13
.86836
8.71648
.38821
8.57508
.40648
8.44889
.48694
8.88180
47
14
.86859
8.71806
.88854
8.57371
.40677
8.41686
.43929
8.88948
46
15
.86898
8.71068
.38888
8.57150
.40911
8.44488
.48968
8.88786
45
16
.80925
8.70819
.38921
8.56088
.40945
8.44880
.48906
8.88670
44
17
.86968
8.70577
.88955
8.56707
.40979
8.44027
.48068
8.88888
48
18
.86991
8.70835
.88988
8.56487
.41018
8.43685
.48067
8.88197
48
19
.87084
8.70094
.89028
8.56866
.41047
8.48688
.48101
8.88018
41
80
.87057
8.09858
.89055
8.56046
.41061
8.43488
.48186
8.81896
40
81
.87090
8.69618
.39089
8.55687
.41115
8.48880
.48170
8.81641
89
88
.8n88
8.60871
.89128
8.56608
.41149
8.48019
.48806
8.81466
88
88
.8n57
8.69181
.89156
8.55880
.41188
8.48819
.48889
8.81871
87
84
.87190
8.68898
.89190
8.55170
.41817
8.48618
.48874
8.81066
86
25
.87888
8.68658
.89888
8.54958
.41851
8.48418
.48806
8.80008
86
86
.87256
8.68414
.88857
8.54784
.41885
8.48818
.48848
8.80718
84
87
.87289
8.68175
.89290
8.54516
.41819
8.48019
.48878
8.80684
88
88
.87328
8.67987
.80824
8.54889
.41858
8.41819
.48418
8.80661
88
89
.87355
8.67700
.89857
8.54062
.41887
3.41680
.48447
8.80107
81
80
.87388
8.67468
.89891
8.58865
.41421
8.41481
.48481
8.80064
80
31
.87488
8.67885
.89425
8.58648
.41455
8.41888
.48516
8.80601
89
88
.87455
8.66089
.89458
8.58488
.41490
8.41085
.48550
8.80619
86
88
.87488
8.66758
.89498
8.58817
.41584
8.40687
.48685
8.89487
87
84
.87581
8.66516
.89686
8.58001
.41558
8.40629
.48680
8.88854
26
85
.87654
8.60281
.89650
8.58766
.41598
8.40488
.48654
8.80O78
86
86
.87688
8.66046
.89598
8.58671
.41086
8.40285
.48660
8.86801
84
87
.87621
8.65811
.88686
8.68887
.41660
8.40088
.48784
8.88710 83
88
.87654
8.65576
.89660
8.58148
.41694
8.80641
.48758
8.88S8B »
89
.87687
2.65842
.89694
8.51929
.41758
8.39045
.48708
8.88848 81
40
.87720
8.65109
.89787
8.51716
.41768
8.89449
.488»
8.88167
80
41
.87754
8.64875
.89761
8.61608
.41797
8.89868
.48868
8.87967
18
42
.87787
8.64012
.89795
8.61889
.41831
8.89068
.48807
8.87806
16
48
.87880
8.64410
.88829
2.51076
.41865
8.88868
.48888
8.87686
17
44
.37853
8.64177
.89662
2.50664
41899
8.88668
.48066
8.87447
16
45
.37887
8.68945
.30696
2.50652
.41983
8.88478
.44001
8.8786J |15|
46
.37980
8.68n4
.89980
2.50440
.41968
8.88879
.44086
8.87068 114
47
.87958
8.68483
.89968
8.50829
.48008
8.88064
.44071
^SS
'48
.87986
8.68252
.89997
8.50018
.48086
8.87891
.44106
8.88780
49
.38020
8.63021
.40031
8.49607
.48070
8.87897
.44140
8.86658
60
.38058
8.68791
.40065
8.49507
.42106
8.87504
.44175
8.86874
51
.88086
8.62561
.40098
8.49886
.48189
8.87811
.44810
8.86106
68
.88180
8.62338
.40182
8.49177
.48178
8.87118
.44844
8.88018
58
.38158
8.62108
.40166
8.48967
.48807
8.86985
.44879
8.85840
54
.38186
2.61874
.40200
8.48758
.42848
8.86733
.44814
8.86668
55
.38280
2.61646
.40234
8.48549
.48876
8.86541
.44840
8.85486
56
.38268
8.61418
.40267
248840
.42310
2.86849
.44884
5SS2
57
8.61190
.40001
2.48188
.42345
8.86158
.44418
8.86188
58
.88320
8.60963
.40835
2.47924
.42379
2.85067
.44458
?-5SK
59
.38358
2.60736
.40809
2.47716
.42418
2.35T76
.44488
884780
60
/
.88886
2.60509
.40408
Ck>tang
6
8.47509 1
.48447
2.35585
.44588
Cotang
8.84094
i
Cotang
Tang
Tang i
Tang
Tang
69- 1
1 67» 1
•^ 10C
c
T^
TABLES,
795
TABLE VU.—ConHnued.
Natural Tangents and Cotangbnts.
t
"0
24- 1
25'» 1
W 1
rr*
§
60
.44528"
Cotanfc
Tang
.46681
Cotang
Tang
Ootang
Tang
Ootang
8.84604
8.14461
.48778
8.06060
.50058
1.96861
1
.44568
8.84428
46666
8.14868
.48800
8.04879
.50969
1.96180
50
s
.44508
8.81858
.46708
8.14186
.48845
8.047%
.51026
1.96979
58
.44627
8.84077
.46787
8.18068
.48881
8.04577
.51068
1.95886
57
.44668
8.88908
.46778
8.18601
.48917
8.04486
.51099
1.96696
56
.44697
8.83787
.46806
8.18689
.48968
8.04876
.51136
1.96557
66
.44788
8.88558
.46648
8.18477
.46989
8.04186
.51173
1.95417
64
.44787
8.28878
.46879
8.18816
.49026
8.06975
.51809
1.05877
58
.44808
8.83804
.46014
8.18154
.49068
8.08895
.51246
1.96187
68
.44887
8.88060
.46950
8.18998
.49096
8.08675
.51868
1.94997
51
.44878
8.88867
.46985
8.18888
.49184
8.08686
.61819
1.94868
50
.44907
8.88688
.47081
8.18071
.40170
8.08876
.51866
1.94718
49
18
.44948
8.88510
.47066
8.18511
.49806
8.06887
.61898
1.94579
46
18
.44977
8.28887
.47098
8.18860
.48848
8.08078
.61480
1.94440
47
14
.45018
8.88164
.47188
8.18190
.408^
8.08889
.61467
1.94801
46
15
.45047
8.81998
.47168
8.18080
.49816
8.08780
.61508
1.94168
45
16
.45088
8.81819
.47199
8.11871
.49651
8.08681
.51540
1.94088
44
17
.45117
8.81647
.47884
8.11711
.49887
8.08488
.51577
1.96666
48
18
.45158
8.81475
.47870
8.11568
.49488
8.08885
.51614
1.99746
48
19
.45187
8.81804
.47806
8.11898
.49459
8.08187
.51661
1.98606
41
90
.46828
8.81188
.47841
8.11888
.49496
8.08089
.61688
1.96470
40
81
.46857
8.80061
.47877
8.11075
.49688
8.01891
.61784
1.93838
89
88
.458^
8.80790
.47418
8.10916
.40306
8.01748
.51761
1.93196
38
88
.45827
8.80619
.47448
8.10756
.49604
8.01596
.51796
1.98067
87
84
.45868
8.80449
.47488
8.10600
.40640
8.01449
.51835
1.98980
36
86
.46897
8.20278
.47519
8.10448
.49677
8.01808
.51878
1.98788
35
86
.45488
8.80106
.47556
8.10884
.49718
8.01155
.51909
1.98646
84
87
.46467
8.19988
.47590
8.10186
.49749
8.O1006
.51946
1.92606
38
88
.45508
8.19789
.47686
8.09969
.49786
8.00668
.51988
1.988n
38
88
.45688
8.19599
.47668
8.09611
.49628
8.00715
.S8020
1.98885
31
80
.46678
8.19480
.47696
8.09654
.49656
8.00660
.52067
1.98096
30
81
.48606
8.19861
.47788
8.09498
.49694
8.00488
.68094
1.91968
89
88
.46648
8.19098
.47769
8.00641
.49081
8.00877
.58181
1.91886
26
88
.46678
8.18088
.47805
8.09184
.49967
8.00181
.68168
1.91690
27
84
.45718
8.18755
.47840
8.09086
.50004
1.99966
.68805
1.91554
86
86
.48748
8.18687
.47876
8.06678
.50040
1.99641
.68848
1.91418
25
86
.46'»4
8.18419
.47918
8.08716
.60076
1.99696
.58879
1.91288
84
87
.46819
8.18851
.47948
8.06560
.50118
1.98660
.68816
1.91147
88
88
.45854
8.18064
.47984
8.06406
.60149
1.99406
.68858
1.91018
28
88
.45889
8.17916
.48019
8.06950
.50185
1.90861
.68880
1.90676
21
40
.46984
8.17749
.48065
8.06094
.50888
1.99116
.68487
1.90741
80
41
.46960
8.17588
.46091
8.07989
.60858
1.96978
.68464
1.90607
48
.46996
8.17416
.48187
•8.07785
.50895
1.96828
.68601
1.90478
48
.46060
8.17!M9
.48168
8.07680
.50881
1.96684
.68586
1.90837
44
.46065
8.17088
.48106
8.07476
.50868
1.96540
.68575
1.90808
46
.46101
8.16917
.48284
8.07881
.50404
1.96396
.68613
1.90069
46
.46186
8.16761
.48270
8.07167
.50441
1.96258
.58650
1.80935
47
.46171
8.16585
.48806
8.07014
.60477
1.96110
.68667
1.89801
48
.46906
8.16480
.48848
8.06660
.60614
1.97966
.58784
1.89667
49
.46848
8.16856
.48878
8.06706
.60560
1.97888
.58761
1.89638
60
.46877
8.16090
.48414
8.06658
.60567
1.97681
.68796
1.89400
51
.46818
8.15985
.48450
8.06400
.60623
1.97588
.58886
1.8926b
58
.46848
8.15760
.48486
8.06847
.50660
1.97895
.52878
1.89188
68
.46888
8.16596
.48581
8.06094
.50696
1.97858
.58910
1.69000
64
.46418
8.15488
.48557
8.06948
.50788
1.97111
.58947
1.66867
66
.46454
8.15868
.48593
8.06790
.60769
1.96969
.58965
1.68784
66
.46489
8.15104
.48689
8.06687
.50606
1.96887
.58088
1.88609
67
.46685
8.14940
.48665
8.05485
.50648
1.96685
.58059
1.88469
68
.46660
8.14777
.48701
8.05888
.50679
1.96544
.58006
1.88887
60
.46696
8.14614
.48787
8.05188
.50916
1.96408
.58184
1.86906
m
.46681
8.14451
.48778
8.05080
.50958
1.96261
.58171
1.88078
#-
Cotang
Tang
Cotangj Tang
Ck>tang
Tang
Ootang
Tang
M* 1
64«
! 63«
68-
796
SURVEYING.
TABLE Vn.—OmHfiued,
Natural Tangents and Cotangents.
"o
%%-
29*»
8a»
ai*
/
Tang-
.5317f
Cotang
Tang
.65481
Cotang
_Tang
1 .67785
Cotang
Tang
.60086
Cotang
1.88078
1.80106
1.78206 1
1.66428
60
1 .63808
1.87W1 1
.55469
1.80881
1 .67774
1.73080
.60126 1.66818 > 50 1
2 .63246
1.87809
.66607
1.80168
I .67818
1.72073
.60165
1.66800 68
8
.63283
1.87677
.65646
1.80064
.67861
1.72867
.60205
1.66000 67
4
.68820
1.87546
.55588
1.79011
.67890
1.72741
j .60d45
1.66000 56
6
53858
1.87415 ,
.65621
1.79788
.67929
1.72625
1 .60284
1.668B1
66
6
.68896
1.87288 '
.65659
1.79665
.57968
1.72609
1 .60324
1.66T78
64
7
.58482
1.8n52 1
.55697
1.79542
.58007
1.72898
, .60364
1.66668
58
8
.53470
1.87021 1
.56736
1.79419
, .58046
1.72278 1! .60403
1.66664
68
9 .58507
1.86891 '
.55774
1.79296
' .58085
1.72163
.60443
1.66446 |61
10 .58M5
1.86760 1
.55812
1.79174
.68124
1.7^047
.60188
1.66887 50
11 .53582
1.86680 1
.55880
1.79061
.68162
1.71982
.60622
1.66288 '49
12{ .53020
l.a<V199 1
.55888
1.78020
1 .68201
1.71817
1.65120 48
13 .53657
1.86869 ,
1.86239 1
.65926
1.78807
, .68240
1.71702
' .60602
1.65011 47
14 .586M
.65964
1.78886
.68279
1.71688
! .60642
1.64008 46
15 .58732
1.86109 1
.56006
1.78668
.68818
1.71478
, .60681
1.64796 45
16 .53769
1.85979 ;
.56041
1.78441
.68357
1.71858
' .60721
1.61687 i44
17 .53807
1.85850 '
.56079
1.78819
< .68896
1.71244
.60761
1.01579 ,43
1.64471 48
18 .53814
1.85720 1
.56117
1.78198
.68435
1.71129
.60801
19, .53882
1.85591 1
.56156
1.78077
.58474
1.71015
.60ftll
1.64868 l41
20 .58920
1.8646S
.66194
1.74056
.58613
1.70901
.60881
1.64256 j40
21 .58957
1.85338
.60232
1.T7834
1 .58552
1.7t)787
.60021
1.64148 30
22 .53995
1.85204
.66270
1.77713
.58591
1.70678
.60960
1.64041 JSS
23 51032
1.85075
.66809
1.77592
.58631
1.70560
.61000
1.680S1
37
24; .54070
1.84946
.66847
1.77471
.58670
1.70446
.61040
1.68686
86
25 .54107
1.84818
.66885
1.77851
.68709
1.70882
.61060
i.68no
85
26
.64145
1.84689 1
.66494
1.77230
; .68748
1.70210
.61180
1.68618
34
27
.541P«l
1.84561
.56462
1.77110
' .68787
1.70106
.61160
1.68506 |S3
28
.61220
1.84488
.56501
1.76990
1 .58826
1.69992
.61800
1.68888 88
29
.54258
1.84806
.6(a539
1.76869
.58866
1.69879
.61240
1.68802
81
80
.54296
1.84177
.56677
1.76740
.68006
1.60766
.61280
1.68186
80
81
.M333
1.84049
.66616
^'W^
.68044
1.69668
.61820
1.68070
80
82
.Msn
1.83022
.56664
1-^
.68988
1.60641
.61360
1.62078
28
33
.M409
1.837M
.56698
i^ASo
.69022
1.69428
.61400
1.62866 27
81
.64446
1.83667
.66781
1.^6871
.69061
1.69316
.61440
1.68r60 ,86
85
64484
1.83540 I
.66760
1.78151
.69101
1.60208
.61480
1.68664
85
36
!m522
1.88413
.66806
1.76Q82
.69140
1.60091
.61520
1.68648
84
37
.61560
1.H3286 i
.66846
1.75913^
i .69179
1.68079
.61661
l.a»142
83
38 .54597
1.83159
.56886
1 75704
1 .60218
1.68866
.61601
1.08886
88
39 .M635
1.83033 ,
.56923
1.75673
1 .69258
1.68764
.61641
1.62280
81
40 .61678
i.aeooe
.56062
1.75666
, .60207
1.68648
.61681
1.62186
80
41 64711
1.82780
.67000 1 1.76487
, .60686
1.68681
« 61781
1.68010
10
421 .54748
1.82654
.67039
1.7581d
.60376
1.68419
.617W
1.61014
18
43 1 .54788
1.82528
.67078
1.75200,
.60416
1.68808
.61801
1.61608
17
44' .54824
1.82402
.67116
1.75082
.60464
1.68106
.61842
1.61708
16
45 .54862
1.82276
.67155 ; 1.74964
.60494
1.68086
.618BS
1.61506
16
46 i .54900
1.82150
.57198 1.74846
1 .60683
1.67974
.61983
1.61408
14
47 .61988
1.82025
.57232 1.747-28
.60578
1.67*03
.61062
1.6188S
18
48 .M9?5
1.81899
.57271 , 1.74610
.50612
1.67752 i
.68008
1.61888
18
49 .66018
1.81774 ,
.57309 , 1.74492
, .59661
1.67641
.62043
1.61170
11
50 .66061
1.81ft49
.57348 1.74375
.59691
1.67530
.62088
1.61074
10
51 ,65089
1.81524 \
.57386 1 1.74257
1 .50730
1.67419
.62124
1.60970
9
52 .55127
1.81399
.57425 1.74140
, .59770 ; 1.67309 I
.62ieM
1.60666
8
53 .55105
1.81274 j
.57464 1.74022
' .59809
1.67198
.68201
i.6om
7
54 .55203
1.81150 '
.57503 1.73905
.59^19
1.67088
.02846
1.00607
6
55; .55241
1.81025 t
.57.541 1.73788
.50888
1.66978
■ ftWfe
1.00668
5
561 .55579
I.H0901
.57580 1 1.73071
.59928
1.66867
1.60419
4
57' .5.ViI7
l.N0r77 1
1.H0653 1
.57619 1.73555
.59967
1.66757
.68866 1.60S4&
3-
58 .55.'r>5
.57657 1.7*438
.60007
1.66647
.68400
lOOEMl
8
59 .5539;i
l.Ha%29
.57«96 1.73321
.60046
1.66538
.68446
1.60187
1
60 .5M.n
CutADK
l.K)4()5
.57735 1 73-.»05
Cotang , Taug
.60086
Cotang
1.66428
.68487
1.0006S
2
Tang
Tang
Cotang,
Tang
1 6
!•
0
0-
59* 1
58«
TABLES.
797
TABLE VU.-^CoHtinued.
Natural Tangents and Cotangents.
I 83» 1
83« 1
84'»
86" 1
Tacj 1
Cotang
Tang
.64941
Cotang
Tang
.67451
Cotang
Tang
.70021
Cotang
1.42815
60
0 .62487
1.60033
1.53986
1.48256
1 .625^7
1.50930
.64962
1.53888
.67493
1.48163
.70064
1.42726
50
S .<)8&68
1.50826
.66024
1.68701
.67536
148070
.70107
1.42638
58
8 .68608
1.50723
.65065
1.58603
.67578
1.47977 i
.701.51
1.42560
57
4 .6S649
1.59620
.66106
1.53505
.67620
1.47886
.70194
1.42462
56
6 .62689
1.59517
.66148
1.53497
.67663
1.47702
.70288
1.42374
55
6 .62730
1.60414
.66189
1.53400
.67706
1.47600
.70281
1.42286
54
7 .62rro
1.50811
.65281
1.63a(^
.67748
1.47807
.70325
1.42198
53
8 .62811
1.50206
.66272
1.68205 ;
1.47514
.70888
1.42110
62
9 .62852
1.50105
.65314
1.58107
.07832
1.47422
.70.112
1.42C22
51
10 .62892
1.50002
.65356
1.53010
.67875
1.47880
.70455
1.41931
60
11 .62933
1.58000
.68397
1.62913
.07917
1.47288
.70409
1.41847
40
12 .62973
1.58797
.66438
1.52816
.67960
1.47146
.70642
1.41750
^
18 .63014
1.58695
.66480
1.52719
.68002
1.47068
.70686
1.41672
47
14 .68055
1.68693
.66621
1.62622
.68045
1.46062
.70629
1.41584
40
16 .68095
1.68490
.66663
1.52525
.08068
1.46870
.70673
1.41497
^
16 .68186
1.58388
.66604
1.62429
.68130 1.46778 1
.70717
1.41400
44
17 .68177
1.68286
.65646
1.62832
.68173
1.46686
.70700
1.41822
48
18 .68217
1.58184
.65688
1.62235
.68215
1.46595
.70804
1 41235
42
19 .68258
1.58083
.65729
1.52139
.CR253
1.4G503
.70*U8
1.41148
41
20 .68290
1.57961
.65771
1.52043
.68301
1.46411
.70691
1.41061
40
21 .68840
1.67879
.65813
1.51946 i
.68»8
1.46320
.70925
1.40974
SO
22 .68880
1.67778
.65854
1.51850
.08386
1.40229
.70979
1 .46 ,37
;j3
28 .68421
1.57676
.65890
1.51754 1
.68429
1.46137
.71023
1.40800
37
24 .68462
1.57575
.65038
1.61658 '
.68471
1.40046
.71066
1.40714
36
25 .63608
1 .57474
.65980
1.61562
.aS514
1.45955
.71110
1.40627
.15
26 .08544
1.57372
.66021
1.51466
.08557
1.45864
.71154
1.40540
4
27 .08584
1.67271
.060G3
1.51370
.68000
1.45773
.71198
1.40454
33
28 .68625
1.6n70
.66105
1.61276
.08642
1.45682
.71242
1.40367
32
29 .63600
1.57069
.66147
1.51179
.C8C85
1.45592
.71285
1 .40281
31
80 .63707
1.56960
.66189
1.51084
.08728
1.45501
.71829
1.40195
30
81 .68748
1.66868
.66280
1.50988
.687n
1.45410
.71878
1.40109
29
32 .G3780
1.567C7
.66272
1.50393
.08814
1.45320
.71417
1.40022
28
88 .63880
1.56667
.66314
1.50797
.08857
1.45229
.71461
1.39936
27
34 .63871
1.56566
.663:^6
1.50702
.0^900
1.45139
.71505
1.89850
26
86 .68912
1.56466
.66398
1.50607
.om2
1.45049
.71540
1.39764
25
86 .68953
1.56306
, .06440
1.60512
.C8D85
1.44958
.71508
1.89679
24
87 .68994
1.66265
.66482
1.5(V417
.09028
1.44808
.71687
1.3^93
23
88 .64036
1.56165
.665^
1.60322
.09071
1.4477B
.71681
1.89507
22
80 .64076
1.66065
.06506
1.50228
.09114
1.44688
.71725
1.39421
21
40 .64117
1.66966
.06608
1.60133
.09157
1.44596
.71760
1.39336
20
41 .64158
1.65866
.60650
1.60038
.09200
1.44506
.71818
1.89250
10
42 .61190
1.55766
.06602
1.49944
.Gin^a
1.44418
.71857
1.39165
IC
43 .64240
1.556G6
.00734
1.49&19
.09-280
1.44329
.71901
1.30079
17
44 .64281
1.55567
.60776
1.49755
.09329
1.44239
.71946
1.88994
16
46 .W822
1.56467
.66818
1.49661
.00372
1.44149
.71990
1.38909 ■•15
46 .M8G8
1.66368
.66860
1.49506
.09416
1.44060
.72034
1.38824 '14
47 1 .&tl04
1.56269
' .60902
1.494?2
.69459
1.43970
.72078
1.88738 ,13
48' .64446
1.55170
.60944
1.4937B
1 .69502
1.43881
.72122
1.38653 ll2
49 .64487
1.55071
1 .6698U
1.492H4
1 .09.M5
1.43792
.72167
1.38568
11
60 .64528
1.64072
1 .67028
1.49190
.09588
1.43703
.72211
1.3}*4M
10
61 .64569
1.64873
' .67071
1.49097
' .09631
1.43614
'' .72256
1.38399
0
52 .64610
1.64774
.67113
1.49003
, .096'. 5
1.43525
.72299
1.38814
8
53. .64652
1.64675
.67155
1.48909
.69718
1.43436
.?2344
1.38229
7
54 .64698
1.54576
.67197
1.48810
.697G1
1.43^7
1 .72388
1.38145
6
55 .64784
1.54478
.67280
1.48722
.69804
l.^-JajS
.72432
1.38060
6
66 .64776
1.64379
.67^82
1.48629
.69^47
1.43109
1 .72477
1.3-976
4
57 .64817
1.M281.
.67824
1.48536
1 .69801
1.43080
1 .72521
1.37891
8
58 .64858
1.541Hf.
.67866
.67400
1.48442
< .69934
1.42992
. .72565
1.37807
2
59 .64399
1 1.64085
1.48349
.69977
1.42903
.72610
1.37722
1
60 64941
1 1.539H6
.67451
Cotang
1.48256
.70021
1.42815
1 .720.-1
1 37038
_0
. 9
^ Cotang
1 Tang
1 Tang
Cotang j Tang
1 Cotang 1 Tang
57»
1 66*
1
56<»
h 1
w
Ogle
798
SURVEYING.
TABLE VU.—ConHnued,
Natural Tangents and Cotangents.
86- 1
87- 1
88* 1
8d*
60
/
Tan^
Cotang
Tang
.75356
Cotang
Tang
Cotang
Tang 1 Cotang
"0
.7a6M
1.87638
1.82704
.78129
1.27994
.Ha+78
1.^490
1
.72609
1.87554
.75401
1.32624
.78175
1.27917
.Hiinrr
1 S*118
GO
2
.72748
1.87470
.75447
1.82544
.78222
1.27841
.Hur75
i.ama
66
3
.72788
1.37886
.75492
1.32464
.78269
1.27764
nil's
1.23^70
67
4
.72882
1.87802
.75588
1.32384
.78316
1.27888
MMl
1.23196
66
5
.72877
1.87218
.75684
1.32804
.78368
1.27611
Hj'ijo
1.S3123
66
6
.72921
1.87134
.75629
1.82224
.78410
1.27585
.81,1*58
1,23050
54
7
.72966
1.87060
.76675
1.82144
.78457
1.27458
K1.SJ6
1.24977
68
8
.78010
1.86967
.75721
1.82064
.78604
1.27382
.Hiri*4
1.2^904
6S
9
.78055
1.86888
.75767
1.81984
.78551
1.27806
HM13
1.3MS31
61
10
.73100
1.86800
.75812
1.31904
.78596
1.27280
MHtl
1.22758
GO
11
.78144
i.8en6
.75858
1.81825
.78645
1.27158
sir.io
1.^<B5
49
\%
.78189
1.86688
.75004
1.81745
.78602
1.27077
.MKV*J
1 .23613
48
18
.78284
1.86540
.75060
1.81666
.78:88
1.27001
>l<^iO
1/^*5S9
47
14
.78878
1.86466
.75090
1.81686
.78786
1.26025
.Mr^^
1.^3467
46
15
.78828
1.86388
.76048
1.81507
.78884
1.26840
.H]Ti;sa
3JE23W
45
16
.78868
1.36800
.76088
1.81427
.78881
1.26774 1
j^irr^
i.saaai
44
17
.78418
1.88217
.76184
1.31348
.78028
1.26696
.t*]KClO
1. 220451
43
18
.78457
1.86184
.76180
1.81209
.78975
1.20622
.81840
1 :?^iru
42
19
.78603
1.36061
.76226
1.81190
.79022
1.26546
,mi^M
1,22104
41
90
.78547
1.85968
.76272
1.81110
.79070
1.26471
>^IW6
i.aaoei
40
21
.78608
1.86885
.78318
1.81061
.79117
1.26395
Sims
i.fitaso
80
22
.73687
1.35802
.76864
1.30052
.79164
1.26319
.HLU{4
1.2t8W
88
98
.78681
1.36719
.76410
1.80878
.79218
1.26244
.^^m
1.^1814
87
24
.78726
1.85637
.76466
1.30796
.79250
1.26160
SJMI
1.S174J
86
25
.78771
1.35554
.70602
i.sono
.79806
1.28098
.SC11*0
1.21670
85
26
.78816
1.35472
.76548
1.80637
.79354
1.20018
-Etij^ia
i.aisos
84
27
.78861
1.86389
.76694
1.30558
.79401
1.25048
.^SSSM
l,i>15i»
83
28
.78906
1.3630r
.76640
1.80480
.79449
1.25867
.82336
1 J214V4
82
20
.78951
1.35221
.76686
1.8(M01
.79496
1.25702
.ft>2»H5
l.Sl38i
81
80
.78996
1.85142
.76788
1.80828
.79544
1.25717
.fs-m
1.21310
80
81
.74041
1.85060
.78779
1.80044
.79601
1.26642
M\^
t.S1238
20
82
.74066
1.84978
.76825
1.80166
.79630
1.25567
.M%^1
1.21166
26
88
.74181
1.84896
.76871
1.80087
.79686
1.26492
^^S^
1 .Si004
27
84
.74176
1.84814
.76918
1.80009
.79734
1.25417
^^tfJO
l.%WA
20
85
.74221
1.34732
.76964
1.29981
.79781
1.25848
,Kvi>78
1.2i]i9&l
25
86
.74267
1.34660
.77010
1.29658
.7B829
1.25868
.N^'7-J7
i.aieTO
24
87
.74812
1.81568
.77057
1.29^/5
.79877
1.25193
Ki-TTfl
1.2G8tT8
23
88
.74857
1.84487
.7ri03
1.29696
.79924
1.25118
,S3SKS
l.iat06
22
89
.74402
1.84406
.Tri49
1.29618
.79972
1.25044
.83*74
l.»»<'^
21
40
.74447
1.84823
.77190
1.29641
.80020
1.24960
.fssam
LaiG43«
20
41
.74498
1.34242
.77242
1.29463
.80067
1.24896
.93^^
1.2063
19
42
.74588
1.84160
.77280
1.29385
.80115
1.24820
K^ha
1 .som
18
48
.74683
1.34079
.77335
1.29307
.80163
1.24746
,?tU,)71
i.aL^3;o
17
44
.74628
1.38998
.77382
1.29229
.80211
1.24672
>.5TJ0
l,2ilW«
16
45
.74674
1.38916
.77428
1.29152
.80258
1.24597
,Knr:9
i.aoa37
15
46
.74719
1.83885
.77475
1.29074
.80006
1.24523
.Kt:ia
1.1^106
14
47
.74764
1.337&4
.77521
1.28907
.80354
1.24449
,Kt,v,e
l.fSOOM
18
48
.74810
1.83678
.77508
1.28919
.80402
1.1M375
.Kmi
J,t5M6t«
18
49
.74855
1.83692
.77615
1.28B42
.80400
1.24801
.Ktm
\.\m^
11
50
.74900
1.33511
.77661
1.28764
.80496
1.24227
.is:Jll5
1J98B2
10
51
.74946
1.83480
.77708
1.28887
.80546
1.24158
.R.*MG5
iJBftll
8
52
.74991
1.33849
.77754
1.28610
.80604
1.24079
.Kl:.l4
1.19740
8
58
.75087
1.88268
.77801
1.28538
.80642
1.24006
.^mi
1.10GC9
7
54
.76082
1.83187
.77848
1.28456
.80690
1.23081
.Mfi33
1J9699
6
55
.75128
1.83107
.77895
1.28379
.80738
1.23858
.8K»3
1.19628
6
56
.75178
1.33026
.7TD41
1.28802
.80786
1.28784
.89718
1.1 9457
4
57
.75219
1.82946
.77988
1.28225
.80834
1.2871G
.83791
1.19887
8
58
.7W64
1.32865
.78035
1.28148
.80882
1.23637
.m^n
1.10316
8
59
.76310
1.82785
.7H0R2
1.28071
.80930
K^l'jQ
M9St46
1
W
.76355
Cotang
1.32704
Tang
.78129
Cotang
1.271K)4
.80978
Cotang
1.28490
1.10175
jO
/
Tang
Tan«
TM«
t
63*
1 62-
1 61-
£
0-
TABLES.
799
TABLE VW^Qmtinued.
Natural Tangents and Cotangents.
9
"o
40*
41- I
42-
48« 1
t
60
Tang ICotang
Tang
.86929
Cotang
Tang 1 CoUug
Tang
Cotang
.88910
1.19175
1.15087
.90040
1.11001
.93258
1.07887
1
.83960
1.19106
.86980
1.14909
.00098
1.10996
.93306
1.07174
59
s
.84009
1.19066
.87031
1.14908
.90146
1.10061
.98360
1.07118
58
8
.84069
1.18964
.87082
1.14884
.90199
1.10667
.9»115
1.07049
57
4
.84106
1.18894
.87188
1.14767
.90261
1.10608
.93469
1.06067
56
ft
.84158
1.18824
.87184
1.14699
.90304
1.10787
.93524
1.06925
55
6
.84206
1.18754
.87236
1.14682
.90357
1.10678
.93578
1.06862
54
7
.84258
1.18664
.87287
1.14566
.90410
1.10607
.98638
1.06600
58
8
.81807
1.18614
.87338
1.14496
.90168
1.10548
.93688
1.06786
68
9
.84367
1.18644
.87389
1.14480
.90516
1.10478
.98742
1.06676
61
10
.UVfl
1.18474
.87441
1.14368
.90669
1.10414
.98797
1.06618
50
11
.84467
1.18404
.87498
1.14296
.90621
1.10849
.98858
1.06651
49
IS
.84607
1.18884
.87M8
1.14229
.90674
1.10285
.98906
1.06489
48
18
.84666
1.18264
.87596
1.14168
.907V7
1.10220
.98961
1.06427
47
14
.84600
1.18194
.87646
1.14096
.90781
1.10156
.94016
1.06866
46
15
.84666
1.18126
.87096
1.14086
.90684
1.10091
.94071
1.06308
45
16
.84706
1.18055
.87749
1.18961
.90887
1.10027
.94125
1.06241
41
17
.84756
1.17866
.87801
1.18894
.90940
1.00063
.94180
1.06179
43
18
.84806
1.17916
.87868
1.18826
.90998
1.09809
.04285
1.06117
42
19
.84856
1.17846
.87904
1.18761
.91046
1.09884
.94290
1.06066
41
20
.84906
1.17777
.87956
1.18604
.91099
1.09770
.94846
1.06994
40
21
.84966
1.17?08
.88007
1.18627
.91158
1.09706
.94400
1.06082
89
22
.86006
1.17638
.88059
1.13561
.91206
1.09642
.94455
1.06870
88
28
.86057
1.17669
.88110
1.18494
.91259
1.09578
.94510
1.05809
87
24
.86107
1.17500
.88102
1.13426
.91818
1.09514
.94565
1.06747
86
26
.85157
1.17480
.88214
1.18361
.91666
1.09450
.94620
1.05685
85
28
.86207
1.17861
.88265
1.18296
.91419
1.09886
.94676
1.06684
84
27
.86257
1.17282
.88817
1.18228
.91478
1.09822
.94781
1.06662
88
28
.86806
1.17228
.88869
1.18162
.91586
1.09268
.94786
1.06601
88
29
.86858
1.17164
.88421
1.13096
.91580
1.09195
.94841
1.05480
81
80
.86406
1.17085
.88478
1.13089
.91688
1.09131
.94896
1.05879
80
81
.85466
1.17016
.88684
1.12963
.01687
1.09067
.94958
1.05817
89
82
.86600
1.16947
.88576
1.12897
.91740
1.C9003
.96007
1.06855
86
88
.86659
1.16878
.88628
1.12881
.91794
1.08040
.95062
1.05194
27
84
.85609
1.16809
.88660
1.12766
.91847
1.08876
.06118
1.05188
86
86
.85660
1.16741
.88788
1.12609
.91901
1.08813
.95178
1.06078
85
86
.85710
1.16672
.88784
1.12688
.91965
1.08749
.96229
1.06010
84
87
.85761
1.16606
.88886
1.12567
.92006
1.06686
.95264
1.04949
88
88
.86811
1.16685
88888
1.12501
.02062
1.06628
.95340
1.04886
88
30
.85862
1.16406
!88940
1.12486
.98116
1.08659
.95805
1.04827
21
40
.86912
1.16896
.88002
1.12860
.92170
1.08496
.95451
1.04766
20
41
.86968
1.16829
.89045
1.12808
.02224
1.06488
.95506
1.04706
19
42
.86014
1.16261
.89097
1.12288
.92277
1.083C9
.95662
1.04644
18
48
.86064
1.16192
.89149
1.12172
.98381
1.08306
.95618
1.01588
17
44
.80115
1.16124
.89201
1.12106
.98385
1.06248
.96673
1.01522
16
45
.86166
1.16056
.89258
1.12041
.98489
1.08179
.95789
1.01461
15
46
.86216
1.15087
.88906
1.11975
.98498
1.08116
.95785
1.01401
14
47
.86267
1.15019
.89858
1.11900
.92547
1.06058
.95841
].OI»IO
18
48
.86818
1.15861
.89410
1.11844
.98601
14)7990
.96897
1.01279
12
49
.86866
1.15788
.80468
1 iirm
.92665
1.07927
.95952
1.01218
11
60
.86419
1.15716
.89515
1.11713
.92709
1.07864
.96008
1.04158
10
61
.86470
1.15647
.89667
1.11648
.92768
1.07801
.96064
1.04097
9
62
.86521
1.15679
.89620
1.11582
.92817
1.07738
.96120
1.O4086
8
68
.86572
1.15511
.89678
1.11517
.92878
1.07876
.96176
1.08976
7
64
.86628
1.15448
.89785
1.11452
.92926
1.07618
.96888
1.08915
6
66
.86674
1.15875
.89777
1.11387
.98960
1.07660
.96888
1.08856
5
66
.86726
1.15806
.89880
1.11821
.93084
1.07487
.96841
1.087M
4
67
.86776
1.15840
.89688
1.11256
.03088
1.07425
.96400
1.08784
8
66
.86627
1.15172
.89935
1.11191
.98143
1.07362
.96457
1.08674
8
69
.88878
1.15104
oggoa
1.11126
.93197
1.07299
.96518
1.08613
1
00
/
.86929
1.15037
!90O4O
1.11061
.93252
Cotang
1.07237
.96669
1.08r63
*
Cotangl Tang
Cotang| Tang
Tang
|Ck>tangj Tang
49*
! 48*
i 47*
1 fc^yLiO
8oo
SURVEYING.
TABLE WU.-^Omtinued,
Natural Tangents and Cotangents.
/
44*
,11,
44- 1.
I
i '
40
,41
42
43
44
45
46
47
48
49
50
51
52
53
55
56
57
58
60
440
/
Tang
Ootang
1
Tang
Cotang I
Tang
Cotang
0
1
2
8
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
80
.96609
.96625
.96681
.96738
.96791
.96a'J0
.96907
.96963
.97020
.97076
.97183
.97189
.9?^46
.97302
.97859
.97416
.97472
.97529
.97586
.97643
.97700
1.08563
1.03493
1.03433
1.03372
1.03812
1.03252
1.03198
1.03132
1.03072
1.0;)012
1.02952
1.08892
1.02832
1.02rr2
1.02713
1.02653
1.02593
1.02533
1 .02474
1.02414
1 02355
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
20
21
22
23
1 34
25
26
27
88
29
30
81
82
83
34
85
86
87
88
39
; 40
.97700
.97756
.97813
.97870
.97927
.979W
.98041
.98098
.98155
.9S^13
.98270
.98327
.98384
.98441
.»MD9
.98556
.98613
.98671
.98728
.98786
.98843
Cotang
1 08355 |40|
1.02295 39
1.08236 881
1.02176 37
1.02117 ,36
1.02057 a">i
1.01998 1*4,
1.01939 33
1.01879 132,
1.01820 31'
1.01761 30 1
1.01702 29
1.016« 128
1.01583 27
1 .01524 1 26 1
1 .01466 1 25
1.01406 IM
1.01847 |23
1.01288 |22
1.01229 |21|
1.01170 |80'
.98843
.98901
.96968
.99016
.990:3
.99131
.99180
.99*17
.99304
.99362
.99420
.99471?
.99536
.99594
.99652
.99710
.99768
.99826
99884
! 99942
1.00000
1.01170
1.01112
1 01053
1.00994
1.00935
1.00876
1.10818
1.00759
1.00701
1.00642
1.00583
1.00525
1.00467
1.0040B
1.00350
1.00291
1.00283
1.00173
1.00116
1.00068
1.00000
90
19
18
17
16
15
14
18
12
11
10
9
8
7
6
5
4
3
8
1
0
/
/
Cotangi Tang
/
Tang 1 ^ 1
/
('otnng i Tung
45»
45- 1 1
46*
Digitized by Cj003I
TABLMS.
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SURVEYING.
TABLE XI.
Volumes by the Prism jid/l Formula. § 320.
i
H BIGHTS.
Correctioai
•rf
for tcniht
inbeigfat
3
1
%
3
4
5
6
7
8
9
10
1
0
1
1
1
2"
2
2
S
8
8
.1
^ i
1 <
1
1
2
;»
3
8
4
5
6
6
.3
0 1
1 8
1
2
3
4
5
6
6
7
8
9
•3
0
4
1
2
4
5
6
7
9
10
11
18
•4
5
—2
- 3
—5
-6
-8
—9
—11
-18
-^14
—16
■ 5
«
2
4
0
7
9
11
13
15
17
19
.6
7
2
4
6
9
11
13
15
17
19
28
-7
8
2
5
7
10
12
J.->
17
20
88
85
A
0
3
6
8
11
14
17
19
82
85
88
•9
1 1
10
8
6
9
12
16
19
82
26
88
81
11
8
7
10
14
17
20
24
27
31
84
.1 1 0
12
4
7
11
15
19
28
26
80
38
87
.» 1 I
18
4
8
12
16
20
24
28
82
86
40
•3 1 1
14
4
9
18
17
22
26
80
85
89
48
•4 *
16
—6
-fl
—14
—19
-83
—88
—82
-87
—42
-46
•5 *
1«
6
10
15
20
S5
80
85
40
44
49
.6 8
17
5
10
16
21
26
81
87
42
47
52
•7 1 '
18
6
11
17
22
28
88
89
44
50
66
.8 1 4
10
6
12
18
28
29
86
41
47
58
59
•9 ^
20
6
12
19
85
81
87
48
49
60
62
21
e
13
19
26
82
89
46
68
68
66
.t 1
22
7
H
20
27
84
41
48
64
61
68
28
7
14
21
28
85
48
&0
57
64
n
24
7
15
22
80
87
44
52
50
67
74
25
-8
-15
—28
-81
-89
-46
-54
-68
-09
-77
20
8
16
21
82
40
48
66
64
78
80
27
8
17
25
88
42
50
68
67
75
88
28
9
17
26
85
43
52
60
69
78
86
29
9
18
27
86
45
51
63
72
81
90
80
9
19
28
87
46
56
66
74
83
98
'
81
10
19
29
88
48
67
67
77
86
96
.1 1 1
82
10
20
80
40
49
69
69
79
89
99
.3
88
10
30
31
41
61
61
71
81
92
102
•3
i 1
84
10
21
81
42
62
63
73
84
04
10-)
• 4
86
-11
-22
—32
-43
-64
-65
-76
-86
-97
-108
• 5
6 1
88
11
22
38
44
66
67
78
89
100
111
•6
87
11
23
31
46
57
69
80
91
103
IH
7
88
12
Z\
35
47
59
70
82
94
106
lit
.8
80
18
24
36
48
60
78
84
96
1(18
1-JO
■9
10
40
12
25
87
49
62
74
86
99
111
m
1
41
13
25
38
51
63
76
89
101
114
127
•Ml
42
13.
26
30
52
65
78
91
101
117
180
.3 « 1
48
13
27
40
53
66
80
98
106
119
133
•3 1 1
44
14
27
41
64
68
81
96
109
122
136
•4 !
45
-14
-28
-42
—66
-69
-83
-97
-111
-125
-139
■5 1 i
' 1 !
40
14
28
48
57
71
85
99
114
128
142
47
15
29
44
68
73
87
102
116
181
145
48
15
30
44
59
74
89
104
119
183
148
■5! Ill
40
15
30
45
60
76
91
106
121
136
151
60
15
81
46
8
62
"4"
77
93
106
188
189
164
1
5
~r"
8
9
10
_ --I
.1
.a
"0"
'3
•4
_5_
■6
.7
.8
.9
Cr>rrfct«oo» for 1
0
1
1
1
1
1
1
lenihi
MOWK
lUL 1
Digitized by
Google
TABLES.
80s
TABLE X\.—ConHnued,
Volumes by the Prismoidal Formula*
1
Heights.
Corrections
for tenths
in height.
1
2
8
4
5
6
7
8
9
10
61
16
31
47
63
79
94
110
126
142
157
.1
2
52
16
82
48
64
80
96
112
1^
144
160
.a
8
58
16
83
49
65
82
08
115
181
147
163
.3
6
51
17
88
50
67
83
100
117
138
150
167
.4
7
65
-17
-81
-51
-68
-86
—102
—119
-186
—163
-170
.5
8
56
17
85
52
69
86
104
12t
138
156
173
.6
10
57
18
86
58
70
88
106
128
141
158
176
:?
12
58
18
86
54
«
90
107
125
143
161
170
14
59
18
86
55
78
91
109
127
148
164
182
■9
15
00
19
87
50
74
98
111
130
148
167
185
1
61
19
88
66
75
94
lis
132
151
169
188
.1
0
62
19
88
57
77
06
115
134
153
1:2
101
.2
4
M
19
30
58
78
91
117
136
156
175
194
.3
0
64
SW
40
50
79
99
119
138
158
178
197
.4
8
05
—JO
-40
-60
-80
-100
-120
-110
-160
—181
-201
• S
10
66
90
41
61
81
102
122
143
163
163
204
.6
12
67
21
41
62
83
103
124
145
165
186
207
.7
14
68
21
42
68
84
105
126
147
168
189
210
.8
:<5
69
21
43
64
85
106
128
149
170
192
213
-9
18
70
24
43
65
86
108
130
151
173
194
216
, 1
71
22
44
66
88
100
131
153
175
197
219
.t
2
73
22
44
67
80
111
138
15G
178
200
222
.a
6
It
2:J
45
68
90
113
185
158
180
208
225
.3
7
74
28
46
69
91
114
137
160
1H3
206
228
.4
9
75
-^
-^
—69
-98
-116
-139
-162
-186
-208
-231
.5
12
76
23
47
70
94
117
141
164
188
211
285
.6
14
77
24
48
71
95
119
143
166
190
214
238
.7
16
78
24
48
72
96
120
144
109
198
217
241
.8
19
70
24
49
73
98
122
no
171
195
219
244
•9
21
80
23
49
74
99
123
148
178
108
222
247
1
81
25
60
75
100
125
150
175
200
225
250
.1
8
8i
2.>
51
76
101
127
152
177
2rtJ
228
258
.a
5
88
26
61
77
102
1-2S
154
179
205
231
266
.3
8
84
26
52
78
104
130
156
181
207
Sf-'JS
259
.4
10
85
—26
-62
—79
-105
-Ml
—157
-184
-2J0
-236
-262
.5
13
86
27
53
80
106
133
159
18B
a 12
2:^9
265
.6
16
87
27
54
81
107
134
161
188
215
242
269
■I
18
88
27
54
81
109
116
103
190
217
244
272
21
80
27
55
82
110
137
165
192
220
247
275
•9
24
00
28
56
83
111
139
167
104
222
250
278
1
91
28
56
81
112
140
169
197
225
258
281
.1
8
02
28
57
85
114
142
170
199
227
256
284
.2
6
98
29
57
86
115
144
172
201
230
2:^
287
.3
9
94
29
58
87
IIG
145
174
2ti3
2J2
261
290
.4
12
95
-29
—69
-88
-117
-147
-176
-2a>
-23.-)
-204
— 293
.5
15
90
30
5U
80
110
148
178
2ur
287
267
296
.6
18
97
80
61)
00
120
150
180
210
240
260
299
.7
21
OS
30
tti)
01
l,'l
151
1HI
212
242
272
802
.8
23
D1
31
61
92
l-Ai
153
1H8
214
244
275
306
•9
26
100
31
6.1
03
3
12:)
154
185
216
247
278
309
1
%
4
•4
5
6
T
8
"JT"
10
. I
.2
.3
•5
.6
•7
^:L.
•9
Corr
tenth
ections for
s in width.
0
0
'
•
1
I
1
1
1
Digitized by
Google
8o6
SURVEYING.
TABLE Xl.—Continued.
Volumes by thb Prismoidal Formula.
1
Hbiohts.
COITBC-
tionsfor
tenths in
1
•
. 11
12
13
14
15
16
17
18
19
20
height.
1
3
4
4
4
6
6
5
6
6
6
,1
0
a
7
7
8
9
9
10
10
11
12
12
.a
0
8
10
11
12
13
14
16
16
17
18
19
• 3
0
4
14
15
16
17
19
20
21
22
23
25
.4
5
—17
—19
—20
—22
—23
—26
—26
—28
—29
—31
• 5
6
20
22
24
26
28
30
31
33
36
37
.6
7
24
26
28
30
32
36
37
39
41
43
.7
8
27
30
32
36
37
40
42
44
47
49
.8
e
31
33
36
39
42
44
47
60
53
66
• 9
10
34
37
40
43
46
49
62
66
59
62
1
11
37
41
44
^
61
64
58
61
65
68
.1
0
12
41
44
48
66
59
63
67
70
74
1
18
44
48
62
66
60
64
68
72
76
80
1
14
48
62
66
60
66
69
73
78
82
86
2
15
—61
—66
—60
—66
—69
—74
—79
—83
—88
—93
, 5
2
16
54
69
64
69
74
79
84
89
94
99
3
17
58
63
68
73
79
84
89
94
100
106
3
18
61
67
72
78
83
89
94
100
106
111
4
19
66
70
76
82
88
94
100
106
111
117
• 9
4
90
68
74
80
86
93
99
106
111
117
123
21
71
78
84
91
97
104
110
117
123
130
.1 1
22
76
81
88
96
102
109
116
122
129
136
If
78
86
92
99
106
114
121
128
135
142
81
89
96
104
111
119
126
133
141
148
25
—86
—93
—100
—108
—116
—123
—131
—139
—147
—164
26
88
96
104
112
120
128
136
144
162
160
27
92
100
108
117
126
133
142
160
168
167
28
96
104
112
121
130
138
147
166
164
173
20
98
107
116
125
134
143
162
161
170
179
30
102
111
120
130
139
148
167
167
176
186
" 1
81
106
116
124
134
144
163
163
172
182
191
1
82
109
119
128
138
148
168
168
178
188
198
2
83
112
122
132
143
163
163
173
183
194
204
• 3
3
84
116
126
136
147
167
168
178
189
199
210
,^
4
35
—119
—130
—140
—151
—162
—173
—184
—194
—205
—216
• 5
5
36
122
133
144
166
167
178
189
200
211
222
.6
6
37
126
137
148
160
171
183
194
206
217
228
,y
8
88
129
141
162
164
176
188
199
211
223
235
.8
9
89
132
144
166
169
181
193
206
217
229
241
• 9
10
40
136
148
160
173
186
198
210
222
236
247
41
139
162
166
177
190
202
216
228
.240
253
.1 I
42
143
166
169
181
194
207
220
233
246
259
.3
3
48
146
169
173
186
199
212
226
239
252
266
• 3
4
44
149
163
177
190
204
217
231
244
268
272
.4
6
45
—163
—167
—181
—194
—208
—222
—236
—260
—264
—278
• 5
7
46
166
170
185
199
213
227
241
266
270
284
.6
8
47
160
174
189
203
218
232
247
261
276
290
.7
10
48
163
178
193
207
222
237
252
267
281
296
.8
11
49
166
181
197
212
227
242
257
272
287
302
• 9
13
50
170
186
201
216
231
247
262
278
293
309
11
12
13
14
15
16
17
18
19
20
.1
.2
3
4
• 5
.6
• 7
.8
• 9
0
1
1
2
2 .
3
3
4
4
tenths in width.
Digitized by
Google
TABLES.
80;
TABLE yH,-— Continued.
Volumes by the Prismoidal Formula.
at
Heights.
Correc-
•0
tions for
tenths in
"^ '
^
11
12
13
14
15
16
17
18
19
20
height.
51
173
189
205
220
236
262
268
283
299
315
.1
2
52
177
193
209
225
241
257
273
289
305
321
.a
3
58
180
196
213
229
246
262
278
294
311
327
• 3
6
54
183
200
217
233
250
287
283
300
817
333
.4
7
55
—187
—204
—221
—238
—266
—272
—289
—306
—323
—340
• 5
8
56
190
207
226
242
259
277
294
311
328
346
.6
10
57
194
211
229
246
264
281
299
317
334
362
.7
12
58
197
215
233
251
269
286
304
322
340
368
.8
14
59
200
219
237
256
273
291
310
328
346
364
• 9
16
60
204
222
241
269
278
296
316
333
362
370
1
61
207
226
245
264
282
301
320
339
358
377
.1
2
62
210
230
249
268
287
306
326
344
364
383
.a
4
63
214
233
253
272
292
311
331
350
369
389
.3
6
64
217
237
257
277
296
316
336
356
375
395
.4
8
65
—221
-241
—261
—281
—301
—321
—341
—361
—381
—401
.5
10
66
224
244
265
286
306
326
346
367
387
407
.6
12
67
227
248
269
290
310
331
352
372
393
414
.7
14
68
231
252
273
294
315
336
357
378
399
420
.8
16
69
234
256
277
598
319
341
362
383
406
426
.9
18
70
238
269
281
302
324
346
367
389
410
432
1
71
241
263
286
307
329
351
373
394
416
438
.1
2
72
244
267
289
311
333
366
378
400
422
444
.a
6
73
248
270
293
316
338
360
383
406
428
451
.3
7
74
251
274
297
320
343
865
388
411
434
467
.4
9
75
—256
—278
—301
—324
—347
—370
—394
—417
—440
—463
• S
12
76
268
281
306
328
362
375
399
422
446
469
.6
14
77
261
286
309
333
356
380
404
428
452
475
.7
16
78
266
289
313
337
361
385
409
433
467
481
.8
19
79
268
293
317
341
366
390
416
439
463
488
.9
21
80
272
296
321
346
370
395
420
444
469
494
1
81
276
300
326
360
376
400
426
460
476
600
.1
3
82
278
304
329
364
380
406
430
456
481
506
.a
6
83
282
307
333
359
384
410
436
461
487
512
.3
8
84
286
311
337
363
389
415
441
467
493
519
.4
10
85
—280
—315
—341
—367
—394
—420
—446
—472
—498
—525
.5
13
86
292
319
346
372
398
425
641
478
504
631
.6
16
87
295
322
349
376
403
430
466
483
610
637
.7
18
88
299
326
363
380
407
435
462
489
616
543
.8
21
89
303
330
367
385
412
440
467
494
622
549
•9
24
90
306
333
361
389
417
444
472
600
628
566
1
91
309
337
366
393
421
449
477
606
634
662
.1
3
92
312
341
369
398
426
464
483
611
640
568
.a
6
93
316
344
373
402
431
469
488
617
545
674
.3
9
94
319
348
377
406
436
464
493
522
551
580
.4
12
95
—323
—362
—381
—410
—440
—469
—498
—528
—557
—586
.5
16
96
326
366
385
415
444
474
604
533
563
693
.6
18
97
329
369
389
419
449
479
609
539
569
699
.7
21
98
333
363
393
423
454
484
614
644
576
606
.8
23
99
336
367
397
428
458
489
619
650
581
611
• 9
26
100
340
370
401
432
463
494
625
566
586
617
11
12
13
14
15
16
17
18
19
20
.1
.2
.3
.4
.5
.6
.7
.8
.9
Correc
tions for
in width.
0
1
1
2
2
3
3
4
4
tenths
Digitized by VjOOQ IC
8o8
SURVEYING,
TABLE XI.— Continued,
Volumes by the Prismoidal Formula.
1
Heights.
Correc-
tions for
•g
1
tenths in
'^
21
22
23
24
25
26
1 27
28
29
30
height.
1
6
7
7
7
8
8
8
9
9
9
.1
0
2
13
14
14
15
15
16
17
17
18
19
.2
0
3
19
20
21
22
23
24
25
28
27
28
• 3
0
4
26
27
28
30
31
32
33
35
36
37
.4
5
—32
—34
—35
—37
—39
—40
—42
—43
—45
—46
.5
6
39
41
43
44
46
48
50
52
64
56
.6
7
45
48
50
62
54
66
58
60
63
66
.7
8
52
54
57
59
62
64
67
69
72
74
.8
9
58
61
64
67
69
72
75
78
81
83
• 9
1 1
10
65
68
71
74
77
80
83
86
90
93
1
11
71
75
78
81
85
88
92
95
98
102
.1
0
12
78
81
85
89
93
96
100
104
107
111
. 3
1
13
84
88
92
96
100
114
108
112
116
120
• 3
1
14
91
95
99
104
108
112
117
121
125
130
.4
2
15
—97
—102
—106
—111
—116
—120
—125
-130
—134
—139
.5
2
16
104
109
114
119
123
128
133
138
143
148
.6
3
17
110
115
121
126
131
136
142
147
152
157
.7
3
18
117
122
128
133
139
144
150
156
161
167
.8
4
19
123
129
135
141
147
152
158
164
170
176
• 9
4
20
130
136
142
148
154
160
167
173
179
185
21
136
143
149
156
162
169
175
181
188
194
.1 1
22
143
149
156
163
170
177
183
190
197
204
.3
2 !
23
149
156
163
170
177
185
192
199
206
213
• 3
f*
24
156
163
170
178
185
193
200
207
216
222
.4
3
25
—162
—170
—177
—185
-193
— 2Q1
—208
—216
—224
—231
• 5
4
26
169
177
185
193
201
209
217
225
233
241
.6
5
27
175
183
192
200
208
217
225
233
242
260
.7
6
28
181
190
199
207
216
225
233
242
251
269
.8
6
29
188
197
206
215
224
233
242
251
260
269
• 9
7
30
194
204
213
222
231
241
250
259
269
278
1
31
201
210
220
230
239
249
258
268
277
287
.1
1
32
207
217
227
237
247
257
267
277
286
296
.2
2
33
214
224
234
244
256
265
275
285
296
306
-3
3
34
220
231
241
252
262
273
283
294
304
316
• 4
4
35
—227
—238
—248
—259
—270
—281
—292
—302
—313
—324
• S
5
36
233
244
256
267
278
289
300
311
322
333
.6
6
37
240
251
263
274
285
297
308
320
331
343
• 7
8
38
246
258
270
281
293
305
317
328
340
362
.8
9
39
253
265
277
289
301
313
325
337
349
361
• 9
10
40
259
272
284
296
309
321
333
346
358
370
1
41
266
278
291
304
316
329
342
354
367
380
. I
1
42
272
285
298
311
324
337
350
363
376
389
. 2
3
43
279
292
305
319
332
345
358
372
386
398
• 3
4
44
285
299
312
326
340
353
367
380
394
407
.4
6
45
—292
—306
—319
—333
-347
—361
—375
—389
—403
—417
• 5
7
46
298
312
327
341 .
355
369
383
398
412
426
.6
8
47
305
319
334
348
363
377
392
406
421
436
7
10
48
311
326
341
356
370
38.5
400
415
430
444
.8
11
49
318
333
348
363
378
393
408
423
439
464
•9
13
50
324
340
355
370
386
401
417
432
448
463
21
22
23
~ir
25
26
27
28
29
30
.1
.2
3
.4
• S
.6
• 7
.8
•9
Correc
tions for
1
2
2
3
4
5
5
6
7
tenths
in width.
Digitized by VjOOQ IC
TABLES.
809
TABLE yL\.— Continued.
V0I.UMES BY THB PrISMOIDAL FORMULA.
2
Heights.
C
/orrec-
1
ti
ons for
•nths in
te
^
21
22
23
24
25
26
27
28
29
30 h
eight.
51
331
346
362
378
394
409
426
441
456
472
I
2
52
337
363
369
385
401
417
433
449
465
481
a
3
53
344
360
376
393
409
425
442
468
474
491 .
3
5
54
350
367
383
400
417
433
450
467
483
500
4
7
55
—356
—373
—390
—407
—424
—441
—468
—475
—492
—509
5
8
56
363
380
398
415
432
449
467
484
601
519 .
6
10
57
369
387
406
422
440
457
476
493
610
528 .
7
12
58
376
394
412
430
448
465
483
601
619
537 .
8
14
59
382
401
419
437
466
473
492
610
628
546
9
15
60
389
407
426
444
463
481
500
519
637
656
1
61
395
414
433
452
471
490
608
627
646
665
1
2
62
402
421
440
459
478
498
517
636
665
674 .
2
4
63
408
428
447
467
486
606
525
544
564
583 .
3
6
64
415
435
454
474
494
514
533
553
673
593 .
4
8
65
—421
—441
—461
—481
—602
—522
—542
—562
—582
—602 .
5
10
(16
428
448
469
489
609
630
650
670
591
611 .
6
12
67
431
455
476
496
617
538
668
579
600
620
7
14
68
441
462
483
504
625
.646
567
688
609
630 .
8
16
69
447
469
490
511
532
554
575
596
618
639 .
9
18
70
454
475
497
519
640
562
583
605
627
648
1
71
460
482
604
526
648
570
592
614
636
657
I
2
72
467
489
611
533
656
578
600
622
644
667
2
6
73
473
496
618
541
563
586
608
631
653
676
3
7
74
480
502
625
548
571
594
617
640
662
685
4
9
75
—486
—509
—532
—566
—579
—601
—626
-648
—671
—694
5
12
76
493
516
640
563
586
610
633
657
680
704
6
14
77
499
523
547
670
694
618
642
666
689
713
7
16
78
506
530
564
578
602
626
650
674
698
722
8
19
79
512
636
561
585
610
634
658
683
707
731
9
21
80
519
643
568
593
617
642
667
691
716
741
1
81
525
560
575
600
625
650
676
700
725
750
I
3
82
531
557
582
607
633
658
683
709
734
759
2
6
83
538
564
589
616
640
666
692
717
743
769
3
8
84
544
670
596
622
648
674
700
726
752
778
4
10
85
—651
—577
—603
—630
—656
—682
—708
—735
—761
—787
5
13
86
657
684
610
637
664
690
717
743
770
796
6
16
87
564
691
618
644
671
698
726
762
779
806
7
18
88
570
698
625
652
679
706
733
760
788
815
8
21
89
577
604
632
659
687
714
742
769
797
824
9
24
90
683
611
639
667
694
722
750
777
806
833
1
91
690
618
646
674
702
730
758
786
815
843
I
3
92
596
625
653
681
710
738
767
795
823
852
2
6
93
603
631
660
689
718
746
775
804
832
861
3
9
94
609
638
667
696
725
754
783
812
841
870
4
12
05
—616
-645
-674
-704
—733
—762
—792
—821
—850
—880
5
15
96
622
652
681
711
741
770
800
830
859
889
6
18
07
629
659
689
719
748
778
808
838
868
898
7
21
98
635
665
696
726
756
786
817
847
877
907
8
23
99
642
672
703
733
764
794
825
856
886
917
9
26
100
648
679
710
741
772
802
833
864
895
926
21
22
23
24
25
26
27
28
29
30
.1
. 3
.3
• 4
.5
.6
.7
.8
•9
Correct!
tenths in
□ns for
width.
1
2
2
3
4
6
5
0
7
Digitized by
Lioogle
8io
SUKP^EYINC,
TABLE XI.— Continued.
Volumes by the Prismoidal Formula.
m
H BIGHTS.
1
Correc-
-5
1
tions for
tenths in
1
31
10
32
10
33
10
34 '
10 1
35
11
36
11
37
11
38
12
30
40
height. 1
12
12
. I
0
2
19
20
20
21
22
22
23
23
124
25
. 2
0
3
29
30
31
31
32
33
34
35
36
37
.3
0
4
38
40
41
42
43
44
46
47
48
49
.4
5
—48
— 49
— 51
—52
—54
—56
—57
—59
—60
—62
• 5
6
57
59
61
63
65
67
68
70
72
74
.6
7
67
69
71
73
76
78
80
82
84
86
.7
8
77
79
81
84
86
89
91
94
96
97
.8
9
86
89
92
94
97
100
103
106
108
HI
• 0
10
96
99
102
105
108
111
114
117
120
123
1
11
105
109
112
115
119
122
126
129
132
136
• T
0
12
115
119
122
126
130
133
137
141
144
148
. 3
1
13
124
128
132
136
140
144
148
152
156
160
•3
1
14
134
138
143
147
151
156
160
164
169
173
• 4
2
15
—144
— 148
— 153
—157
-162
—167
—171
—176
— 181
— 185
.5
2
16
153
158
163
168
173
178
183
188
193
198
.6
3
17
163
168
173
178
183
189
194
199
205
210
.7
3
18
172
178
183
189
194
200
206
211
217
222
.8
4
19
182
188
194
199
205
211
217
223
229
235
.9
4
20
191
198
204
210
216
222
228
235
241
247
1
21
201
207
214
220
227
233
240
246
253
259
. I
1
22
210
217
224
231
238
244
251
258
265
272
. 3
2
23
220
227
234
241
248
256
263
270
277
284
.3
2
24
230
237
244
252
259
267
274
281
289
296
.4
3
25
-233
-247
-255
—262
—270
-278
—285
—293
—301
—309
• 5
4
26
?49
257
265
273
281
289
297
305
313
321
.6
5
27
258
267
275
283
292
300
308
317
325
333
• 7
5
28
268
277
285
294
302
311
320
328
337
346
.8
6
20
277
286
295
304
313
322
331
340
349
358
• 9
7 1
30
287
296
306
315
324
333
343
352
361
370
1
31
297
306
316
325
3.35
344
354
364
373
383
.1
1
32
306
316
326
336
346
356
365
375
385
395
.3
2
33
316
326
336
346
356
367
377
387
397
407
.3
3
34
325
336
346
357
367
378
388
399
409
420
• 4
4
35
—335
—346
— 356
— 367
—378
—389
—400
—410
—421
—432
.5
5
36
344
356
367
378
389
400
411
422
433
444
.6
6
37
354
365
377
388
400
411
423
434
445
457
• 7
8
38
364
375
387
399
410
422
434
446
457
469
.8
0
39
373
385
397
409
421
433
445
457
469
481
• 9
10
40
383
395
407
420
432
444
457
469
481
494
1
41
392
405
418
430
443
456
468
481
494
606
.1
1
•2
402
415
428
441
454
467
480
493
506
519
.3
3
43
411
425
438
451
465
478
491
504
518
631
• 3
4
44
421
435
448
462
475
489
502
516
530
543
•4
6
45
—431
—444
—458
—472
—486
—500
—514
—528
—542
—556
5
7
46
440
454
469
483
497
511
525
540
554
568
.6
8
47
450
464
479
493
508
522
537
651
566
580
.7
10
48
459
474
489
504
519
533
548
563
578
593
.8
11
49
469
484
499
514
529
544
560
575
590
605
• 9
13
50
478
494
509
525
540
556
571
586
602
617
31
32
33
34
35
36
37
38
39
40
. 1
. 3
• 3
.4
• S
.6
.7
.8
• 9
ContK
tenths
rtionsfor
in width.
1
2
3
4
5
6
8
9
10
Digitized by CjjOOQIC
TABLES.
8ll
TABLE XL— Can/mwei.
Volumes by thb Prismoidal Formula.
CA
Heights.
Correc-
•2
tions for
tenths in
^
31
32
33
34
35
36
37
38
39
40
height.
51
488
604
519
535
551
567
582
598
614
630
. I
2
52
498
ol4
530
546
562
578
594
610
626
642
. 2
3
53
507
623
540
656
573
589
605
622
638
654
• 3
5
54
517
633
650
567
583
600
617
633
650
667
•4
7
55
—526
—643
—660
—577
—694
—611
—628
—645
—662
—679
• 5
8
56
536
563
570
588
606
622
640
657
674
691
.6
10
57
645
663
581
598
616
633
651
669
686
704
• 7
12
58
555
573
591
609
627
644
662
680
698
716
.8
14
50
565
583
601
619
637
656
674
692
710
728
• 0
15
60
674
693
611
630
648
667
685
704
722
741
1
61
584
602
621
640
659
678
697
715
734
763
.1
2
62
693
612
631
661
670
689
708
727
746
765
.3
4
63
603
622
642
661
681
700
719
739
758
778
• 3
6
64
612
632
652
672
691
711
731
751
770
790
4
8
65
—622
—642
—662
—682
—702
—722
—742
— 7G2
—782
—802
• 5
10
66
631
662
672
693
713
733
7.54
774
794
816
.6
12
67
641
662
682
703
724
744
765
786
806
827
• 7
14
68
661
672
693
714
736
756
777
798
819
840
.8
16
60
660
681
703
724
745
767
;88
809
831
862
9
18
70
670
691
713
736
756
778
7\*9
821
843
864
1
71
679
701
723
746
767
789
811
833
855
877
.1
2
72
689
711
733
756
778
8(;o
822
844
867
889
. 2
5
73
698
721
744
766
789
811
834
856
879
901
.3
7
74
708
731
764
777
799
822
845
868
891
914
■ 4
9
75
—718
—741
—764
—787
—810
—833
—856
—880
—903
—926
•5
12
76
727
751
774
798
821
844
868
891
915
938
.6
14
77
737
760
784
808
832
856
879
903
927
961
•7
16
78
746
770
794
819
843
867
891
915
939
963
.8
19
70
756
780
806
829
853
878
902
927
951
975
9
21
80
765
790
816
840
864
889
914
938
963
988
1
81
776
800
826
850
875
900
925
950
976
lOCO
3
8?
785
810
835
860
886
911
936
962
987
1012
. 2
6
83
794
820
846
871
897
922
948
973
999
1025
.3
8
84
804
830
856
881
907
933
959
985
1011
1037
•4
10
85
—813
—840
—866
—892
—918
—944
—971
-997
— 1023
—1049
5
13
86
823
849
876
902
929
956
982
1009
1035
1062
.6
16
87
832
859
886
913
940
967
994
1020
1047
1074
• 7
18
88
842
869
896
923
951
978
1005
1032
1069
1086
.8
21
80
852
879
906
934
961
989
1016
1044
1071
1098
9
24
00
861
889
917
944
972
1000
1028
1056
1083
nil
1
01
871
899
927
955
983
1011
1039
106V
1096
1123
.1
3
02
880
909
937
965
994
1022
1051
1079
1107
1136
. 2
6
03
890
919
947
976
1006
1033
1062
1091
1119
1148
• 3
9
04
899
928
957
986
1015
1041
1073
1102
1131
1160
.4
12
05
—909
—933
—968
—997
—1026
— lOoO
— 1085
—1114
—1144
—1173
.5
15
06
919
948
978
1007
1037
1067
1096
1126
1156
1185
.6
18
97
928
968
988
1018
1048
107S
1108
1138
1168
1198
.7
21
98
938
968
998
1028
1059
1089
1119
1149
1180
1210
.8
23
00
947
978
1008
1039
1069
1100
1131
1161
1192
1222
• 9
26
100
957
988
1019
1049
1080
nil
36
1142
1173
1204
1235
31
32
33
• 3
34
35
• 5
37
38
39
40
. I
.2
• 4
.6
• 7
.8
• 9
tions for
in width.
1
2
3
4
5
6
8
9
10
tenths
Digitized by VjOOQ IC
8l2
SUKVEYING.
TABLE XI. ^Continued.
V0I.UMES BY THE PrISMOIDAL FORMULA.
2
Hbights.
Correc-
tions for
I
tenths in
41
42
48
44
45
46
47
48
49
50
height.
1
13
13
13
14
14
14
15
15
15
15
. r
0
2
25
26
27
27
28
28
29
30
30
31
.3
0
3
38
39
40
41
42
43
44
44
45
46
.3
0
4
61
52
53
54
56
57
58
59
60
62
4
5
—63
—66
—66
—68
—69
—71
—73
—74
—76
—77
• S
6
76
78
80
81
83
85
87
89
91
93
.6
7
89
91
93
95
97
99
102
104
106
108
.7
8
101
104
106
109
111
114
116
119
121
123
.8
0
114
117
119
122
125
128
131
133
136
139
• 9
10
127
130
133
136
139
142
145
148
151
154
1
11
139
143
146
149
153
156
160
163
166
170
. I
0
12
152
156
159
163
167
170
174
178
181
185
.2
1
13
165
169
173
177
181
185
189
193
197
201
3
1
14
177
181
186
190
194
199
203
207
212
216
.4
2
15
—190
—194
— 199
—204
—208
—213
—218
—222
—227
—231
.5
2
16
203
207
212
217
222
227
232
237
242
247
.6
3
17
215
220
226
231
236
241
247
252
257
262
.7
3
18
228
233
239
244
250
256
261
267
272
278
.8
4
19
240
246
262
268
264
270
276
281
287
293
•9
4
20
253
259
265
272
278
284
290
296
302
309
1
21
266
272
279
285
292
298
305
311
318
324
. I
1
22
278
285
292
299
306
312
319
326
333
340
.3
2
23
291
298
305
312
319
327
334
341
348
355
.3
2
24
304
311
319
326
333
3^11
348
356
363
370
.4
3
25
—316
—324
—332
—340
—347
—355
—363
—370
—378
—386
.5
4
26
329
337
346
363
361
369
377
385
393
401
.6
5
27
342
350
3.68
367
376
:«3
392
400
408
417
.7
5
28
354
363
372
380
389
398
406
415
423
432
.8
6
29
367
376
386
394
403
412
421
430
439
448
.9
7
30
380
389
398
407
417
426
435
444
454
463
1
31
392
402
411
421
431
440
450
459
469
478
. I
1
32
405
415
425
436
444
454
464
474
484
494
.3
2
33
418
428
438
448
468
469
479
489
499
509
.3
3
34
430
441
451
462
472
483
493
.504
614
525
•4
4
35
—443
—464
-466
—476
—486
—497
—508
—619
—629
—540
• 5
5
36
456
467
478
489
600
511
622
533
544
556
.6
6
37
468
480
491
502
614
525
637
548
660
671
.7
8
38
481
493
504
516
528
540
551
563
575
586
.8
a
39
494
506
518
530
542
564
566
578
590
602
• 9
10
40
506
519
531
543
556
568
580
593
605
617
1
41
519
631
544
557
569
582
595
607
620
633
. I
1
42
531
544
557
570
583
596
609
622
635
648
.2
3
43
544
657
571
684
597
610
624
637
650
664
.3
4
44
557
670
584
598
611
626
638
652
665
679
• 4
6
45
—569
— 5r;3
—697
—611
—626
—639
—653
—667
—681
—694
• 5
7
46
682
596
610
626
639
653
667
681
696
m
.6
8
47
696
609
624
638
663
667
682
696
711
.7
10
48
607
622
637
652
667
681
696
711
726
741
.8
11
49
620
636
650
665
681
696 I
710
726
741
756
• 9
13 1
50
633
648
664
679
694
710
726
741
756
772
41
42
43
44
45
.5
46
47
48
49
50
.,
. 3
.3
• 4
.6
.7
.8
• 9
tions for
in width.
1
3
' 1
6
7
8
10
11
13
tenths
Digitized by VjOOQ IC
TABLES.
813
TABLE yi\-^Conlinutd.
Volumes bv thb Prismoidal Formula.
ui
Heights.
Correc-
"0
tions for
tenths in
5
41
42
43
44
45
46
724
47
48
49
50
height.
Al
645
661
677
693
708
740
756
771
787
.1
2
52
658
674
690
703
706
722
738
754
770
786
802
. 3
3
53
671
687
720
736
762
768
785
802
818
• 3
5
54
683
700
717
733
750
767
783
800
817
833
.4
7
55
—696
—713
—730
—747
—764
—781
—798
—815
—832
—849
• S
8
56
709
726
743
760
778
795
812
830
847
864
.6
10
57
721
739
756
774
792
809
827
844
862
880
.7
12
58
734
752
770
788
806
823
841
859
877
895
.8
14
59
747
765
783
801
819
833
856
874
892
910
• 9
16
60
759
778
790
815
833
852
870
889
907
926
1
61
772
791
810
828
847
866
885
994
923
941
.1
2
62
785
804
823
842
801
880
899
919
938
957
.2
4
63
797
817
836
856
875
894
914
933
953
972
• 3
6
64
810
830
849
869
889
909
928
948
968
988
• 4
8
65
—823
—843
—863
—883
—903
—923
—943
—963
—983
—1003
• 5
10
66
835
856
876
896
917
937
957
978
998
1019
.6
12
67
848
869
889
910
931
951
972
993
1013
1034
.7
14
68
860
881
902
923
944
965
986
1007
1028
1049
.8
16
69
873
894
916
937
958
980
1001
1022
1044
1066
• 9
18
70
886
907
929
951
972
994
1015
1037
1069
1080
1
71
898
920
942
964
986
1008
1030
1052
1074
1096
.1
2
72
911
933
956
978
1000
1022
1044
1067
1089
nil
.a
3
73
924
946
969
991
1014
1036
1059
1081
1104
1127
.3
7
74
936
959
982
1005
1028
1051
1073
1096
1119
1142
.4
9
75
—949
—972
—995
—1019
— 1042
—1065
—1088
—nil
—1134
—1157
.5
12
76
962
985
1009
1032
1056
1079
1102
1126
1149
1173
.6
14
77
974
998
1022
1046
1069
1093
1117
1141
1165
1188
.7
16
78
987
1011
1035
1059
1083
1107
1131
1156
1180
1204
.8
19
79
1000
1024
lOlH
1073
1097
1122
1146
1170
1195
1219
• 9
21
80
1012
1037
1062
1086
1111
1136
1160
1185
1210
1236
1
81
1025
1050
1075
1100
1125
1150
1175
1200
1225
1250
.1
3
82
1038
1063
1088
• 1114
1139
1164
1190
1215
1240
1265
.3
6
83
1050
1076
1102
1127
1153
1178
1204
1230
1255
1281
• 3
8
84
1063
1089
1115
1141
1167
1193
1219
1244
1270
1296
.4
10
85
—1076
—1102
— 1128
— 11.54
—1181
—1207
— 1233
—1259
—1285
—1312
.5
13
86
1088
1115
1141
1168
1194
1221
1248
1274
1301
1327
.6
16
87
1101
1128
1155
1181
1208
12:?5
1262
1289
1316
1343
.7
18
88
1114
1141
1168
1195
1222
1240
1277
1304
1331
1358
.8
21
89
1126
1154
1181
1209
1236
1204
1291
1319
1346
1373
• 9
24
90
1139
1167
1194
1222
1250
1278
1306
1333
1361
1389
91
1152
1180
1208
1236
1264
1292
1320
1348
1376
1404
.1 3
92
1164
1193
1221
1249
1278
13(;6
1335
1303
1391
1420
. 2
6
93
1177
120G
1234
1263
1232
1320
1340
1378
1406
1435
•3
9
94
1190
1219
1248
1277
1306
1335
1364
1393
1J22
1451
•4
12
95
— 1202
-1231
-1261
-1200
— 1319
— 1319
—1378
— 1407
— 1437
— 1466
•S
15
96
1215
1244
1274
1304
1333
13G3
1393
1422
1452
1481
.6
18
97
1227
^1257
1287
1317
1347
1377
1407
1437
1467
1497
.7
21
98
1240
'1270
1301
1331
1361
1391
1422
1452
1482
1512
.8
23
99
1253
1283
1314
1344
1375
1406
1436
1467
1497
1528
-9
26
lOOv
1265
41
1296
42
1327
43
1358
1389
1420
1451
1481
1512
1543
44
45
46
47
48
49
50
.1
.2
.:s
•4
.5
.6
.7
.8
.9
...
— —
Correc
tions for
1
3
4
6
7
8
10
11
13
tenths
in width.
Digitized by VjOOQ IC
8i4
SURVEYING.
Table XII. — Azimdths op Polaris
The Star and the Azimuth are W. of N. when the hour angle is Itsn
The Ahoument is the star's hour angle (or 23 h. 56 min.
To Find the True Meridian the azimuth must be laid off to the ea^ when the
&
%
Azimuths for Latitude —
Date.
1901.
•H
^
i
^
6
0
0
0
0
0
0
0
0
0
0
0
la
^
0)
30
32
34
36
38
40
42
44
46
48
50
h.
m.
m.
m.
m.
m.
/
/
/
1
/
/
/
'
/
/
/
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Jan. 1
4.
5
5
6
5
2.
2
2
2
2
2
2
2
2
2.
2.
15
9.
9.
9.
9.
10
3.
3.
3.
4
4
4
4
4.
4.
4.
6
Feb. 1
15
14
14.
14.
14.
14
5.
6.
5.
6.
6
6
6
6.
6.
7
7
19
19
19
19.
19.
7
7
7.
7.
8
8
8.
8.
9
9
9.
23.
24
24
24.
24.
9
9
9
9.
9.
10
10.
10.
11
11.
12
Mar. 1
28.
28.
29
29
29.
10.
U
11
11.
11.
12
12.
13
13.
14
14.
15
33
33.
34
34
34.
12.
12.
13
13
13.
14
14.
15
16.
16
17
Apr. 1
15
•
38
38.
38.
39
39.
14
14.
14.
15
15.
16
16.
17
18
18.
19
42.
43
43.
44
44.
16
16
16.
17
17.
18
18.
19.
20
21
21.
47.
48
48.
49
49
17.
18
18.
19
19.
20
20.
21.
22.
23
24
May 1
52.
63
53.
54
54
19.
20
20.
21
21.
2?
22.
23.
24.
26.
26.
15
JL
-Si
58
68.
59
59.
21.
23
21.
23.
22
24
22.
24.
23.
26
M
26
25
27
26
28
27
29
28
30
29
31
June 1
1
T^
^^^
^"^
15
7
7.
%\
9
9!
25
25.
26
26.
27
28
29
30
31.
32.
34
July 1
15
12
13
13.
14
15
27
27
27.
28.
29
30
31
32
33.
35
36.
17
18
18.
19
20
28.
29
29.
30
31
32
33
34
36
37.
39
22
23
23.
24.
25.
30.
31
31.
32
33
34
35
36.
38
39.
41.
Aug. 1
27
28
29
29.
30.
32
32.
33
34
35
36
37
38.
40
42
43.
15
32.
33.
34
35
36
33.
34.
35
36
37
38
39.
41
42.
44
46
Sept. 1
15
37.
38.
39.
40.
41.
35.
36
37
38
39
40
41.
43
44.
46.
48.
43
44
45
46
47
37
38
38.
39
40.
42
43.
45
46.
48.
60.
48
49
50.
51.
62.
39
39.
40.
41.
42.
44
45.
47
49
51
53
Oct. 1
53.
59 1
1 54.
_55.
57
58
40.
42.
41.
43
42.
44
43.
45.
44.
46.
46
48
47.
49.
49
51.
61
53.
53
65.
65
57.
15
•^
T
T
Nov 1
2
4
6
7.
s!
9.
44
45
46
47
48.
50
61.
63.
65.
67.rHri
15
lo"
11
12.
14
16.
46
47
48
49
50.
52
53.
55.
6^ Ml 1 A9 1
16
21.
27.
17
23
29
18.
24.
30.
20
26
32.
21
28
34
47.
49.
51
48.
50.
52
49.
51.
53.
51
53
55
52.
54.
56.
54
56
68 1
56
58
62
57.'
■ST
62
Mbtr
62
64.
62
64.
66.
64
67
69.
Dec. 1
15
33.
35
37
38.
40.
53
54
55.
56.
62
^(T
64
66.
69
72
Tabular
40
41.
48
43.
60
45
52
47
54
54.
56
57.
58.
62
64
64
66
66
68.
68.
71
71.
73.
74.
76
46
66.
57.
59
63
"ST
/>ai^.
52.
54.
56.
-S^i
8
68
59.
62.
64.
64
66
66
68
68
70
70.
72.
73
75
76
78
79
81.
1
^^a
T
^^T
^
2
3
4
5
6
3
6^
8.
\\
13
15.
61.
63
64.
66
68
70
72
74.
77.
80.
84
13.
16
18.
21
23.
63.
65
66.
68
70
72
74.
77
79.
82.
86
21
23.
26
23
31.
65
66.
68
70
72
74
76.
79
82
86
88.
28.
31.
34.
37
40.
67
68.
70
72
74
76
78.
81
84
87
91
37
39.
43
46
49.
69
70.
72
74
76
78
80.
83
86
89.
93.
7
8
9
10
11
45.
64.
48.
58
52
55
59
70.
72.
72
74
74
76
75.
77.
77.
79.
80
82
82.
84.
85.
87.
88.
90.
91.
94
95.
08
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12!
16!
21
74
76
77.
79.
81.
84
86.
89.
92.
96
100.
15
19.
24
29
34.
76
77.
79.
81.
83.
86
88.
91.
96
98.
103
12
27
40.
67.
■ST
32
46.
37.
53
43.
50
77.
79.
81.
83
79.
81.
83
85
81.
83
85
87
83.
85
87
89
85.
87.
89.
91.
88
90
92
94
90.
93
95
97
94
96
98
100
97
99.
101
103
105
107.
110
112
13
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32
TT
14
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101. 105.
103. 107.
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I.,
TABLES.
815
FOR ALL Houa Anolxh. S 881a.
than 11*" 58" and E. of N. when the hour an^Ie is greater than 11^ 68".
minus the star's hour anf?le), for the years Kiv<^°*
hour angle is less than 11^ SS", and tb the west when it is greater than 11>> 58".
Time of
Culmina-
tion after
mean
noon.
i
S
m
h.
i
m.
s
1^
i
m.
i
m.
0
SO
•
88
84
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0
86
0
88
•
40
•
48
•
44
/
0
46
0
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e
60
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m.
m.
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/
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8 39.8
6
28
52.
1
■»■
83
81.
79.
78
85
83
81.
79.
87
85
83
81.
89
87
86
83.
91
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87.
85.
90
1^
93
^90.
100
98
96
98.
103.
101.
99.
97
107.
1C5
103
100.
112.
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109.
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77.
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88.
88.
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98.
102
2 46.9
46.
42.
88.
84.
29.
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76
77.
79.
81.
84
86.
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92.
96
100
1 51.8
%
56.
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45
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72.
74
76
77.
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82
84.
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94
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78
82.
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86
83.
31
28.
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65
66.
68
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82.
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20 41.6
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63
64
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71
15 47.9
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INDEX.
PAGI
Abney Level and Clinometer 141
Accuracy of the Stadia Method 279
Attainable by Steel Tapes, and Metallic Wires in Measurements. 521
Adjustments, Method of Studying 4
■General Principle of Reversion 15
of Compass 15
of Level 63
Precise 607, 70&
of Plane Table 119
of Sextant ill
of Solar Compass 41
Attachment • • loa
of Transit 86
of Angles in Triangulation Systems • 539
Triangle 541
Quadrilateral 542
Larger Systems 554
of Polygonal Systems in Leveling 613
Agreement, Want of, between Surveyors 437 ^
Alignment, Corrections for, in Base-line Measurements 510
to invisible Stations 480
Altitude of a Heavenly Body 583
Aneroid Barometer 127
Angle Measurement in Triangulation 525 to 536, 705
Angles Measured by Chain 12
Angular Measurements in Subdivision 404
Areas of Cross-sections in Rivers , 306
819
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820 INDEX.
PACB
Areas of Land 187
by Triangular Subdivision 188
from Boundary Lines 189
from Rectangular Coordinates of the Corners 209
of Irregular Figures 216
Formulae for Derived 685
Azimuth Defined 11
and Latitude by Observations on Circumpolar Stars. . ..558, 568, 707
of Polaris at Elongation, Table of 33
of Polaris at Any Hour 569
Balancing a Survey 198
Barometer, Aneroid 127
Use of the Aneroid 136
Barometric Formulae Derived 128
Tables 133
Base-line and its Connections 475
Measurement 495, 513, 706
Broken, Reduction to a Straight Line 516
Reduction to Sea Level 516
Computation of Unmeasured Portion 520
Summary of Corrections to 517
Base-lines on the U. S. Surveys, List of 702
Bed Ownership in Water Fronts 640
Bench-marks 74, 307
in Cities 42S
in Triangulation 493, 614, 707
Bc-row-pits 468
Boundaries, Identification of 229
Water Boundaries 232
Bubble, Value of one Division of 58, 604
Bubbles, Level 55
Construction of Tube 56
Propositions Concerning 56, 57
Use of, in Measuring Small Vertical Angles 58
Angular Value of one Division found in three ways 58, 708
General Considerations 59
Buoys and Buoy Flags 299
Catenary Effect with Steel Tapes 4J0. 513
Chain, Engineer's 5
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INDEX. • 8jI
VAQB
Chain, Gunter's 5
Erroneous Lengths of 6
Testing of 6
Permanent Provision for 7
Standard Temperature 7
Use of 8
On Level Ground 3
On Uneven Ground 3
Number and Use of Pins 9
Exercises with 11
Chaining over a Hill il
Across a Valley 11
Random Lines • 11
Check Readings in Topographical Surveying 269
Circumpolar Stars, Times of Elongation and Culmination 560
Pole Distances of " 561
Azimuth of Polaris at Elongation 33
City Surveying 400
Land Surveying Methods Inadequate.. • 400
The Transit *'•., 401
The Steel Tape 401
Laying out a Town Site 403
Provision for Growth 403
Contour Maps 404, 415
Angular Measurements in Subdivision 404
Laying out the Ground 405
Plat to be Geometrically Consistent 407
Monuments : 407
Surveys for Subdivision 409
Datum Plane 413
Location of Streets 413
Sewer Systsms 414
Water Supply 414
Methods of Measurement 415
Retracing Lines 415
Erroneous Standards 416
True Standards 41?
Use of Tape 418
Normal Tension •••• 420
Working Tension 424
Effect of Wind 435
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822 INDEX.
FAGB
City Suryeying — Effect of Slope • 426
Temperature Correction 426
Checks 427
Miscellaneous Problems 428
Improvement of Streets 428
Permanent Bench-marks 428
Value of an Existing Monument 429
Significance of Possession 431, 636
Disturbed Corners and Inconsistent Plats 432
Surplus and Deficiency 433» 638
Investigation and Interpretation of Deeds 435
OflSce Records 435
Preservation of Lines 436
Want of Agreement between Surveyors 437
Clinometer
(Coefficient of Expansion of Steel Tapes 504
of Brass Wires 504
Compass Level — The Architect's 690
Compass, Needle. Description of 13
Adjustments 15
Use of A. 34
Setting of the Declination 36
Local Attractions, Sources of 36
Tests of 37
To Establish a Line of a Given Bearing. 37
To Find a True Bearing of a Line 37
To Retrace ap Old Line 37
Exerciits-gHyita' Compass and Chain 38
Compass, Prismatic Pocket 38
Compass, Solar 39
Adjustments of 41
Use of 44
Finding the Declination of the Sua 44
Errors in Azimuth due to Errors in Declination and Latitude. 49
Table of Such Errors 51
Time of Day Suitable for Observations 52
Exercises with the Solar Compass 53
Convergence of Meridians 179, 628
Contour Lines, Propositions Concerning 275
found by Transit and Stadia 275
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INDEX. 823
PAG£
Contour Lines, Found by Clinometer 391
Used in Computing Earth-work 423
Contour Maps in City Work 404, 415
Coordinate Protractor 168
Corner Monuments in Land Surveying 181, 634
Cannot be Established by Surveyors 635
Cross-sectioning in Earth-work 452
Cross-section Polar Protractor 114
Cross-sections, Areas of, in Rivers 306 ■
of Least Resistance 344
in Earth-work 450
Cross-wires, Illumination of 568
Setting of 252
Current-meters 316
Rating of 3I7.-33I
Use of, in Streams 316, 331
Conduits 329
Curvature and Refraction, Tables of 481, 599
Datum Planes in Cities 413
Declination of Magnetic Needle 20
Variations in 20
The Daily Variation 20
The Secular Variation 21
Other Variations 29
To Find the Declination with the Compass and an Observation
on Polaris r 29
Lines of Equal Declination in United States orlSTfJJontc Lines.. 23
Formulae for Finding the Declination at 82 Points in the United
States and Canada 25
Declination of the Sun 44
Method of Finding .•« 44
Correction for Refraction 1 45
Table of Corrections 48
Deeds, Investigation and Interpretation of 229, 435
Deficiency, Treatment of, in City Work 433, 638
Differences, Finite, Method of 685
Construction of Tables 685
Derivation of Formulae for Evaluating Irregular Areas 688
Direction Meter 332
Discharge of Streams 310
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824 INDEX.
pAcm
Discharge of Streams — Measuring Mean Velocities of Water Currents. 310
Submerged Floats 311
Current Meter 316
Rating the Meter 317
Rod Floats 323
Comparison of Methods 324
Relative Rates of Flow in Different Parts of the Cross-section. . 325
Computation of the Mean Velocity over the Cross-section 328
Sub-currents 332
Flow over Weirs ^ 332
Formuls and Corrections 335
Miner's Inch 338
Flow of Water in Open Channels, Formulae for 339
Kutter*s Formulae 342
Formulae for Brick Conduits 343
Cross-sections of Least Resistance 344
Sediment Observations 345
Collecting the Specimens 347
Measuring out the Samples 347
Siphoning off, Filtering, Weighing, etc 348
Disturbed Corners 432
Dredging 469
Earth-work, see Volumes.
Earth-work Tables ^ 456
Elevation of Stations in Triangulation 480
Elongation of Polaris, Times of 32
Error, ProporHonaie 2
Errors, Compensating and Cumulative 2
in Precise Leveling 612
Estimates, Preliminary, in Earth-work 443^ 46$
Excavations under Water 469
Excess, Spherical 542
Expansion, Coefficient of 504
Field Notes, Changes in 3
in Land Surveying 190
in Differential Leveling 75
in Profile Leveling 78
in Topographical Surveying , ••265, 715
Filar Micrometer •^^^^^^^^*^^•. 52S
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INDEX. 825
PACB
Floats, Submerge^ 311
Flow of Water in Open Channels 339
in Brick Conduits 343
Cross-sections of Least Resistance 344
(See also Discharge of Streams.)
Gauge , Hook 335
Water 307, 617
Geodetic Leveling, Trigonometrical and Spirit 592 to 617
Geodetic Positions, Computation of 587
Derivation of Formulae for 691
Geodetic Surveying 472
Triangulation Systems 473
Base-line and its Connections 475, 706
Reconnaissance 477
Instrumental Outfit for 479
Direction of Invisible Stations 480
Heights of Stations 480
Construction of Stations 485
Targets 486
Heliotropes 490
Station Marks 492
Measurement of Base-lines 495, 706
Use of Steel Tape in 497
Method of Mounting and Stretching 498
M. jaderin's Method 501
Absolute Length 503
Coefficient of Expansion 504
Modulus of Elasticity 505
Effect of Sag 505
Temperature Correction 507
with Metallic Thermometer 508
Correction for Alignment 510
Sag 513
Pull 513
Reduction of Broken Base to a Straight Line 516
Reduction to Sea Level 516
Summary of Corrections 517
Computation of an Unmeasured Portion 520
Accuracy attainable with Steel Tapes and Metallic Wires 521
Measurement of the Angles 525, 705
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S20
INDEX.
PACB
Geodetic Surveying — Instruments " 525
Filar Micrometer 528
Programme of Observations 531
Repeating Method 532
Continuous Reading around the Horizon 533, 705
Atmospheric Conditions 535
Geodetic Night Signals 536
Reduction to the Centre 536
Adjustment of the Measured Angles 539
Equations of Condition 539
Adjustment of a Triangle 541
Spherical Excess 542
Adjustment of a Quadrilateral 542
Geometrical Conditions 542
Angle Equation Adjustment 542
Side Equation Adjustment 545
Rigorous Adjustment for Angle and Side Equation 549
Example 552
Adjustment of Larger Systems 554
Computing the Sides of the Triangles 554
Latitude and Azimuth 558
Conditions of the Discussion 558
Found by Observations on Circumpolar Stars at Elongation
and Culmination 558, 707
Observation for Latitude, Two Methods 562
Correction to the Meridian 564
Observation for Azimuth 565
Correction to Elongation 567
The Target 56S
Illumination of Cross-wires 568
Time and Longitude 571
Fundamental Relations 571
Sidereal to Mean Time 575
Mean to Sidereal Time 576
Change from Sidereal to Mean Time 577
Observation for Time 578
Selection of Stars 578
List of Southern Time Stars 580
Mean Time of Transit 582
Altitude of Star 583
Making the Observations 584
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INDEX. 827
PAGB
Geodetic Surveying — Programme of Obscnrations 586
Computing the Geodetic Positions 587
Table oi L M Z Coefficients 589
Example 591
Geodetic Leveling 592, 708
Trigonometrical Leveling 592
Formulx for Reciprocal Observations. 593
Observations at one Station only 595
an observed Angle of Depression 597
Value of the Coefficient of Refraction 698
Precise Spirit-leveling 590, 708
Instruments 600
Instrumental Constants 604, 708
Daily Adjustments 607, 710
Field Methods 609, 712
Limits of Error 612, 715
Adjustment of Polygonal Systems 613
Determination of the Elevation of Mean Tide 617
Grade, Leveling for 81
Grading over Extended Surfaces 440
Hand Level. Locke's 81
Heights of Stations in Triangulation .. 480
Heliotropes 490
Hook Gauge 335
Horizontal Angle Measurement 93
Hydrographic Surveying 293, 715, 720
Location of Soundings 294
Two Angles read on Shore 295
in the Boat 295, 721
One Range' and One Angle 298
Buoys, Buoy Flags, and Range Poles 299
One Range and Time Intervals 200
Intersecting Ranges 300
Cords or Wires 300
Making the Soundings 301, 721
Lead 301
Line 301
Sounding Poles 303
Soundings in Running Water 303
Water-surface Plane of Reference 303
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828 INDEX.
PAGS
Hydrographic Surveying — Lines of Equal Depth 304
Soundings on fixed Cross sections in Rivers 304
Soundings for the Study of Sand-waves 305
Areas of Cross-section 306
Bench-marks 307
Water-gauges 307
Water-levels 308
River Slope .*.... 309
Finding the Discharge of Streams (see Discharge of Streams) . 310
Illumination of Cross-wires 568
Inaccessible Object.
Distance to and Elevation of . . .■ 105
Length and Bearing of a Line joining two such 107
Integrations with Current Meter 317
Isogonic Lines in the United States 33 and PI. II.
Judicial Functions of Surveyors 633
Kutter's Formul.€ 34a
Lakes, Riparian Rights in 232 and 641
Land Monuments 173
Land Surveying 172
Laying out Land 172
Land Monuments 173
Significance and A uthority of Monuments 174
Lost Monuments 175
United States Method 176
Origin of 176
Reference Lines 177
Division into Townships 178
Division into Sections 17S
Convergence of Meridians 179
Corner Monuments i8r
The Subdivision of Sections 183
The Running of Parallels 185
Areas of Land 187
by Triangular Subdivision 1S8
by use of Chain alone 188
by use of Compass or Transit and Chain 189
Digitized byVjOOQlC
INDEX. 829
PACT
Land Surveying by use of Transit and Stadia 189
from Bearing and Length of Boundary Lines 189
Field Notes 190
Computing the Area 193
The Method Stated 193
Latitudes, Departures and Meridian Distances 193
Computing Latitudes and Departures 195
Balancing 19S
Rules for Balancing 200
Error of Closure 201
Form of Reduction 202
Area Correction Due to Erroneous Length of Chain 205
From Rectangular Coordinates of Corners 208
Conditions of Application 208
Method Stated 209
Form of Reduction 2H
Supplying Missing or Erroneous Data 2H
Bearing and Length of One Course unknown 213
Bearing of One Course and Length of Another unknown 213
Two Bearings unknown 214
Lengths of Two Courses unknown 214
Plotting the Field Notes 216
Areas of Irregular Figures 216
Offsets at Irregular Intervals 216
Regular Intervals • 218
Subdivision of Land 221
To cut oflf by a Line through a given Point 223
in a given Direction 223
Principles and Laws Governing Resurveys 228
Interpretation of Deeds and Identification of Boundaries 229
Water Boundaries and Meandered Lines 232
Surplus and Deficiency 233
Exercises 234
Latitude, Geocentric and Geodetic 691
Latitude and Azimuth 558
Laws Governing Land Surveys 228
Leads used in Soundings 301
Length, Standards of 416
Absolute, of Steel Tapes 503
Lettering on Maps 629
Level Bubbles 55
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8jO INDEX.
PACK
Level» Hand 8i
Leveling, Ordinary 71
Precise Spirit 599, 708
Trigonometric 592
Leveling Rods 70
Levels, Water 308
Level Surface 55
Level, The Engineer's 60
The Architect's 69a
Adjustments 63
Relative Importance of 68
Focussing and Parallax 68
Use of the Level 71
Back-and Fore-sights 71
Differential Leveling 72
Length of Sights 73
Bench-marks 74, 75
The Record 75. 76
The Field Work 76
Profile Leveling 77
The Record 78
Leveling for Grade 81
Exercises 82
Level Trier 59
Line, Sounding 301, 721
Lines, Clearing out • 480
Preservation of, in Cities 436
L. M. Z. Coefficients, Table of 589
Formulae Derived 691
Location of Railroad on Map 287
Longitude, Determination of 571
Map Lettering 629
Maps in Topographical Survey 278, 722
in Railroad Surveying 283
Projection of 618
Meander Lines, Extension of, in Boundaries 232, 638
Mean Tide, Water, Determination of Elevation of 617
Mean Velocities of Water Currents 310
Measurements of Volume 438
Meridians, Convergence of 176, 628
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INDEX, 831
PAOB
Meridians, Principal, used on the U. S. Surveys 702
Metallic Thermometer Temperature Corrections 508
Micrometer, Filar 528
Mineral Surveyors, Instructions to 643
Mining Surveying 349
Mining Claims, Title to 349
Location Surveys 351
Lode Claims, Surveys of 351
Patent Surveying 355
Placer Claims 368
Mill Sites 368
Amended Surveys 368
Adverse Surveys 369
Underground Surveying 370
Carrying the Meridian into the Mine 380
Underground Leveling 389
Mapping the Survey 390
Problems of Underground Surveying 392
Surface Surveys 397
Court Maps 398
Missing Data, Supplying of 211
Bearing and Length of One Course unknown 213
of One Course and Length of Another unknown 213
Two Bearings unknown 214
Two Lengths unknown 214
Modulus of Elasticity of Steel Tape 505
Monuments 173
at Section Corners 181
in City Work 407. 489, 432
in Triangulation 492, 707
Significance and Authority of 174
Lost 175
Night Signals in Triangulation 536
Normal Tension of Tape in City Work 420
Odometer 139
Office Records 435
Optical Square 142
Parallax, How Removed 68
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832 . INDEX.
PACK
Parallel Ruler 169
Parallels of Latitude, how run l8s
Pantograph, Theory of i6x
Varieties 164
Use of 165
Pedometer 137
Pivot Correction in Leveling 605, 709
Plane Table 117
Adjustments 119
Use of 120
Location by Resection 123
Resection on Three Points 123
Resection on Two Points 124
Use of Stadia 125
Exercises 126
Planimeters 143
Theory of the Polar Planimeter 144
To Find Length of Arm 150
Suspended Planimeter 152
Rolling Planimeter 152
Theory of 154
Test of Accuracy of Planimeter Measurements 157
Use of the Planimeter 158
Accuracy of Planimeter Measurements 160
Used in Computing Earth-work 465
Plats, to be Geometrically Consistent 407
Inconsistent 432
Michigan Regulations Concerning 731
Plotting in Land Surveying 216
Topographical Surveying 268, 270, 722
Railroad Surveying 285
Plumb-line, its great Utility 55
Use of, in Chaining 9
Deviations of • 56
Polaris, Times of Elongation of • • • 32
Azimuth of, at Elongation ... . 33
Azimuth of, at any Hour, Table XII 814
Porro's Telescope 249
Possession, Significance of •....431, 636
Preservation of Lines 436
Prismoid, The Warped Surface 454
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IN£>JSX. 833
PACK
Prismoid — The Henck*s 461
Prismoidal Forms 448
Formulae 448
Tables 456
Precise Spirit Level 599, 708
Projection of Maps 618
Rectangular Projection 618
Trapezoidal Projection 619
Simple Conic Projection 620
De risle's Conic Projection 621
Bonne's Conic Projection 621
Polyconic Projection 622
Formulx used in 622
Derivation of Formulae 691
Table of Constants 625, 765
Summary 626
Convergence of Meridians 628
Protractors 166
Three-armed 167
Paper Protractor , 167
Co5rdinate 168
Topographical 271, 273
Cross-section Polar 114
Public Lands in the United States 176
(See also Land Surveying.)
Railroad Topographical Surveying 281
Objects of the Survey 281
Field Work 281
Another Method 291
Maps 283
Plotting the Survey 285
Making the Location on the Map 287
Ranges and Range Poles in Sounding 300
Reconnaissance in Triangulation 477 to 492
Records, Office, in City Work 435
Reduction to the Center in Triangulation 536
Refraction and Curvature, Table of Values of 481, 597
Refraction, Table of Mean Values 563
Tabular Corrections to Declination for, with Solar Compass. ... 47
in Trigonometrical Leveling 592
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834 mDEx.
PACK
Refraction — Coefficient of 598
Repeating, Method in Triangulation 532
Results, Number of Significant Figures in 3
Resurveys, Principles and Laws Concerning 228
Retracing Lines in City Work 415
Riparian Rights in Water Fronts 638 to 641
River Slope 309
Rod Floats 323
Ruler, Parallel 169
Sag Effect with Steel Tapes 505, 513
Sand Waves, Study of ! 305
Scales 169
Sections in Land Surveying 175 to 178
Sediment Observations 345
Sewer Systems 414
Sextant ". 108
Theory no
Adjustments in
Use 112, 295. 721
Exercises 112
Wood's Double 113
Shrinkage of Earth-work 468
Sides, Computation of, in Triangulation 554
Simpson's Rules Derived 690
Slope of River Surface 309
Solar Attachments 99
The Saegmuller Attachment 102
Adjustments 102
Solar Compass (see Compass, Solar).
Soundings, Location of 294, 721
Making 301, 721
Spherical Excess 542
Stadia Methods, Accuracy of 279
Stadia Rod, Graduation of » 253
Stadia Surveying (see Topographical Surveying).
Standards of Length in City Work 416, 417
Stars for Time Determinations. List of 5S0
Circumpolar, Times of Elongation and Culmination of 560
Pole Distances of 563
Stations, Direction of Invisible 480
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IN'DEX. 835
PAGB
Stations — Heights of in Triangulation • 480
Construction in Triangulation 485
Marks at in Triangulation 492
Steps, Length of Men's 138
Steel Tapes 9
in City Work 401, 416 to 427
in Base-line Measurement 497 to 5 13
Straight Lines Run by Transit 95
Streams, Discharge of 310
Streets, Location 413
Improvement of 428
Stretch of Steel Tapes 513
Subdivision of Land 221
Cutting off by a Line from a given Point 221
in a given Direction 223
Subdivision of Town Plats 409
Submerged Floats .' ." 311
Surplus, Treatment of, in City Work 233, 433, 638
Surveying Land (see Land Surveying).
Surveyors, Want of Agreement 437
Judicial Functions of 633
Cannot Change Original Monuments 634
The Location of Lost Monuments 634
Re-location of Extinct Interior Corners 634
Cannot * * Establish " Corners 635
Significance of Possession 636
Surplus and Deficiency 638
Meander Lines, Extension 638
Meander Lines, not Boundary Lines 638
Extension of Water Fronts 639
• Bed Ownership in Water Fronts 640
Riparian Rights in Small Lakes 641
Tables, Construction of 685
List of :
I. Trigonometric Formulae 753
II. For Converting Meters, Feet, and Chains 757
III. Logarithms of Numbers to Four Places 758
IIIa. Logarithms of Numbers and Trigonometrical Functions to
Four Places 760
IV. Logarithmic Traverse Tables, Four Places 764
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836 INDEX.
PAGK
Table list of : V. Stadia Tables 772
VI. Natural Sines and Cosines 780
VII. Natural Tangents and Cotangents 789
VIII. Coordinates in Polyconic Projections 801
IX. Value of Coefl5cient C in Kutter's Formulae 802
X. Diameters of Brick Conduits 803
XI. Volumes by the Prismoidal Formulae 804
XII. Azimuths of Polaris for all Hours 814
Tape, Steel (see Steel Tape).
Targets in Triangulation 486
Temperature Correction in Tapes 507
Tension of Tape in City Work 420, 424
Tide Water, Determination of Elevation of Mean 617
Time and Longitude, Determination of 571 to 586
Time, Sidereal and Mean 575
Time Stars, List of 580
Three-point Problem, Four Solutions 296
Topographical Surveying 237
Transit and Stadia Method 238
Fundamental Relations 238
The Use of an Interval Factor 244
Systematic Errors in Stadia Measurements 245^
Adaptation to Inclined Sights 246
The Porro Telescope 249
Setting the Cross-wires 252
Graduating the Stadia Rod 253
The Topography 257
Field Work 257, 717
Reduction of Notes 265
Plotting the Stadia Li ne 268
Side Readings 270
Check Readings 269
Contour Lines . . *. 275
The Final Map 278
Topographical Symbols 279
Accuracy of the Stadia Method 280
Topographical Symbols 279, 630
Topography, Railroad (see R. R. Topographical Surveying).
Townships in Land Surveying 178
Town Site, Laying Out 403
Transit, The Engineer's 83
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INDEX. 837
PAGI
Transit — General Description 83
Adjustments 86
Relative Importance of Adjustments 89
Eccentricity in Horizontal Circle •• 90
Inclination of Vertical Axis * 91
Horizontal Axis 92
Collimation Error 93
Use of the Transit 93
Measurement of Horizontal Angles 93
Vertical Angles 94
Running out Straight Lines 95
Traversing 97
The Solar Attachment 99
Adjustments of Saegmuller Attachment 102
The Gradienter Attachment 104
Care of the Transit 104
Exercises 105
Transit in City Work 401
in Mining Work
in Topographical Work 252
Triangulation, Instruments Used in 525
Programme of Observations 531
Adjustment of Angles 539
Computing Sides 554
Latitude and Azimuth 558
Time and Longitude 571
Computation of Geodetic Positions 587
Triangulation Systems 473
Traversing 97
Trigonometer (see Coordinate Protractor).
Trigonometrical Leveling 592
Formula 707
Variation of Magnetic Needle (see Declination).
Velocities of Water Flow 310
in Vertical Planes 317, 326
in Horizontal Planes 325
Verniers 18
The Smallest Reading of 20
Rule for Reading 20
Vertical Angle Measurement ••••• 94
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«38 INDEX.
rACB
Volumes, Measurement of 438
The Elementary Form 438
Grading over Extended Surfaces 440
Approximate Estimates by Means of Contours 443
Prismoid '. 448
Prismoidal Formula 44S
Areas of Cross-section 450
Center and Side Heights 451
Area of Three-level Sections 451
Cross-sectioning 452
The Warped Surface Prismoid 454
Construction of Tables for 456
The Henck Prismoid 461
Comparison of the Henck and Warped Surface Prismoid \. 463
Preliminary Estimates from the Profile 465
Borrow Pits 468
Shrinkage of Earth-work., 468
Excavations under Water 469
Water Boundaries 232
Water Currents, Mean Velocity of 310
Sub-surface 332
Water Fronts, Riparian Rights in 638
Water Gauges 307
Water Levels 308
Water Supply, Surveys for 414
Weddel's Rule Derived 690
Weirs, Flow Over 332
Formulae and Corrections 335
Wire Interval. Value of 606
Wood's Double Sextant 113
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Armsby't Manual of Cattle-feedinc xamo,
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Biidd and Hansen's American Horticultaral Ma,niialt
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Part n. — Systematic Pomology xamo,
Downing's Fruits and Fruit-trees of America 8vo,
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Sanderson's Insects Injurious to Staple Crops. xamo,
Insects Injurious to Garden Crops, (/n preparation,)
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SiDckbridge's Rocks and Soils. 8to,
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Compound Rlreted Girders as Applied in BoUdiags 8to,
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Skeleton Construction In BuikUngs 8to,
Briggs** Modem American School BniMings. Sro,
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Theatre Fires and Panics lamo,
Molly's Carpeatais' and Joiners! Handbook x8mo,
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Siebert and Biggin's Modem Stone-cutting and Masonry .Sro, i &•
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Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches.
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Wood's Rustless Coatings: Corrosion and Blsctrolysis of Iron and Steel... S?o, 400
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lamot morocco, i 9^
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ICBsff's Manval of Aasayinc lamo, t 99
O'DriscolTs Notes on the Treatment of Gold Ores 8?o» a o#
Rkkstta and Miller's Notes on Assaying 8?o, s o#
Ulke'a Modem Blectrolytic Copper Refining 8?o, 3 oo
Wilson's Cyanide Processes lamo, 1 tfa
Chlorination Procsss « lamo, i 9*
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Craig's Azimuth 4to, 3 9*
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Helm's Principles of Mathematical Chemistry. (Morgan.) xaino» i so
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Hopkins's OU-chemists' Handbook 8vo, 3 00
Jackson's Directions for Laboratory Work in Physiological Chemistry. .8to, x as
Keep's Cast Iron 8vo, a 50
Ladd's Manual of (^lantitaOte Chemical Analysis xamo, x 00
Umdauer's Spectrum Analysis. (Tingle.) 8to, 3 00
Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) xamo. x 00
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Lodge's Notes on Assaying and Metalhargical Laboratory Expsfimaiili. (/n
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Mason's Water-supply. (Considered Principally from a Sanitary Standpo^)
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Matthews's The Textile Fibres. (/« pret«.)
Meyer's Determination of Radicles in Carbon Compounds. (Tingle.) . . xamo, x 00
Miller's Manual of Assaying »««o. « •o
Milter's Elementary Text-book of Chemistry xamo. x 90
Morgan's Outline of Theory of Sotation and its Results tamo, x 00
Blements of Physical Chemistry lamo, a 00
Morsel Calculations used hi Cane-sugar Factories i6mo, morocco, x so
Mttlliken's General Method for the Identification of Pure Organic Compounds.
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Ifichols's Water-supply. (Considered mainly from a (^lemical and Sanitary
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O'Brine's Laboratory Guide in Chemical Analyab Svo, a o*
O'Driscoll's Notes on the Treatment of (JoM Ores 8to, a om
Ost and Kolbeck's Text-book of Chemical Technology. (Lorenx— Boxart)
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Pinner's Introduction to Organic Chemistry. (Austen.) xamo, i 5*
Poole's Calorific Power of Fuels 8to, 3 00
Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
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RIckktta and Miller's Notes on Assaying Syo, 3 o«
Rldeal's Sewage and the Bacterial Purification of Sewage 8to, 3 99
Disinfection and the Presemition of Food. 8to, 4 00
Riggs's Elementary Manual for the Chemical Laboratory 8to, x as
Ruddiraan's Incompatibilities in Prescriptions. 8to, a o»
Sabin's Industrial and Artistic Technology of Paints and Varnish 8to, 3 00
Salkowski's Physiok>gical and Pathok>gical Chemistry. (Omdortf.). . . .8to, a 50
SchimpTs Text-book of Volumetric Analysis xamo, a so
Essentials of Vohimetric Analysis xamo, i ag
Spencer's Handbook for Chemists of Beet-sugar Houses. x6mo, morocco, 3 oo
Handbook for Sugar Manufacturers and their Chemists. . x6mo, moroccot a 00
Slockbridge's Rocks and Soils 8to, a so
* Tillman's Elementary Lessons In Heat 8to, x so
* Descriptire (General Chemistry 8to» 3 oo
Treadwell's QualiUtiTe Analysis. (HalL) 8to, 3 00
QuantiUtiTe Analysis. (HalL). 8to» 4 00
Tttmeaure and Russell's Public Water-supplies 8?o, 5 o*
Van Derenter's Physical Chemistry for B^^inners. (Boltwood.) xamo, x SO
* Walke's Lectures on EzplosiTes 8to, 4 00
Washington's Manual of the Chemical Analysis of Rocks. (In prett.)
Wassermann's Immune Sera: H«emolysins, Cytotozins, and Precipitins. (Bol-
duan.) xamo, x 00
Wells's Laboratory Guide in QualiUtire Chemical Analysis. 8to, x so
Short Course in Inorganic QualitatiTa Chemical Analysis for Engineering
Students xamo, x so
Whipple's Microscopy of Drinking-water 8vo, 3 so
Wiechmann's Sugar Analysis Small 8to. a so
Wilson's Cyanide Processes. xamo, x 5«
Chlorination Process xamo. x SO
WuUing's Elementary Course in Inorganic Phaimaceutical and Medical (Siem-
istry xamo, a 00
CIVIL ElfGINESRIlfG.
BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF BNOnfEERIHO
RAILWAY ENGINEBRING.
Baker's Engineers' Surveyinf Instruments ■. lamc, 3 ••
Bizby's Graphical Computing Table Paper xpiXa4i inches. as
00 Burr's Ancient and Modem Bngineerinc and the Isthmian Cand. (Postage,
a? cents addltionaL) 8to, net, 3 so
Comstock's Field Astronomy for Engineers. 8to, a so
Davis's Elevation and Stadia Tables 8vo, x oo
BUotf s Engineering for Land Drainage xamo, x so
Practical Farm Drainage xamo, x 00
fohrelTs Sewerage. (Designing and Maintenance.) Sve, 300
Ireltag's Architectural Engineering, ad Edition Rewritten Sve, 3 $&
Digitized by VjOOQ IC
Frtnch and Itm** Stereotomy Bro, a 9»
Ooodhua*! ManiciiMl ImproTvmMiti lamo, z 70
Ooodrich'i Economic DiipoMl of Towns' RefiiM 8to, s 9*
Oore'i Elementi of Oeodcty 8to, 2 50
Hsyford'i Text-book of Oeodetie Aitronomy 8to, 5 00
Horing't Readj Ref eranco Tabkt (Conronion Facton) i6mo, morocco, 2 90
Howo't Retaining Walk for Bartb xamo, i as
Johnson's Theory and Practice of Siurejinc Small 8to, 4 00
Statics by Algebraic and Graphic Methods 8vo. a 00
Kiersted's Sewage Disposal tamo, i as
Laplace's Philosophical Essay on ProbaUllties. (Tniscott and Emory.) zamo, a oo
Mahan's Treatise on Ciril Engineering. (z873 ) (Wood.) 8to» s oo
^ DescriptiTe Geometry 8?o, z s*
Uerriman's BlementB of Predse Sorreying and Geodesy 8v0» a 50
Elements of Sanitary Engineering 8?o, a 00
Uerriman and Brooks's Handbook for Sunreyors i6mo» morocco* a oo
Hagenf s Plans Sonreying 8vo> 3 50
Ogden's Sewer Design. lamo, a oe
Patton's Treatise on CiTil Engineering 8to half leather, 9 9*
Reed** Topographical Drawing and Sketching 4to« 8 00
RideaTs Sewage and the Bacterial Pmification of Sewage 8to, 3 go
Siebert and Biggin's Modem Stone-cutting and Masonry Sro, z 90
Smith's Manual of Topographical Drawixig. (McMillan.) 8vo, a 0»
Sondericker's (haphie Statics* wnn Applications to Trasses, Beams, and
Arches. 8vo, a 00
Tiaylor and Thompson's Treatise on Concrete, Plain and Reinforced, (/n prett. )
* Trantwine's CiTil Engineer's Pockets-book i6mo, morocco, s 00
Wait's Engineering and Architectural Jurisprudence 8to, 6 00
Sheep, 6 so
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8to, s 00
Shoop, 5 90
Law of Contracts Sro, 3 00
Warren's Stereotomy— Problems in Stone-cutting. 8to, a 9*
Webb's Problems in the Ufe and Adjustment of Engineering Instruments.
zomo, morocco, z as
* Wheeler's Elementary Coarse of QtU Engineering Sto, 4 00
Wilson's Topographic Sunreying Sto, 3 90
BRIDGES AKD ROOFS.
Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8to, a 00
* Thames RItct Bridge 4to, paper. 5 00
Burr's Course on the Otresies in Bridges and Roof Trusses, Arched Ribs, and
Suspension Bridges. 8to, 3 9»
Du Bois*s Mechanics of Engineering. VoL n Small 4to, zo 00
Poster's Treatise on Wooden Trestle Bridges 4to, s 00
Vowler's Coffer-dam Process for Piers 8to, a 90
9rsone's Roof TTusMs 8to, z as
Bridge Trusses 8to, a 9»
Arches in Wood, Iron, and Stone 8tOi a 9*
Howe's Treatise on Arches 8to, 4 oe
Design of Simple Roof •'trasses in Wood and Steel 8to, a 00
Johnson, Bryan, and Tumeaure's Theory and Practice in the Designing of
Modem Framed Stractures. . Small 4to, zo 00
Merrlman and Jacoby's Text-book on Roofs and Bridges:
Part I. —Stresses in Simple Trusses 8to, a 90
Part n.— Oraphic Statics 8to, a 90
Part in.— Bridge Design. 4th Edition, Rewritten Sto, a 90
Part IV.—Higher Structures. Sto, a 90
Morfson's Memphis Bridge 4to, zo 00
6
Digitized by VjOOQ IC
Wadd»irt Dt Pwtlbut, • Podwt'boftk far Bridg» Fngjimn. . . ittmo, mofocea, j oo
SpMiflntiontforStMlBridgat 121110, i sg
W«o4'tTrMitiMOiitb«Th«oryof thtConftrnctionof BridgwandRoofi.STo, a oo
Wllglifs Dtrigniag of Dnw-tpftnt:
Put I. — Plat«-glrd«r Dnwt 8to, a 90
Part IL— RlTeted-truM and Pin-connacted Long-apan Diaiia 8to» s 9*
Two partB In ona Toloma 8to» 3 9a
HTBRAUUCS.
Bailn't Biparimanti apoa tha Contractkm of tha Liquid Vein laatting from an
Orifica. (TTavtwina.) 8to, 2 oa
Ba«af*a Traatiaa on Hfdfanlca. 8to, 5 00
Charah't Machanica U Bnginaerinc 8to, 6 oa
Diaffimma of Maan VtlocitT of Watar in Opan Channalt ptpar, i 5»
Cafln'to Graphical Solntian of Hydraolk Problams ^. . . .iteio, morocao, a 5»
flatbar'a Dynamomatan, and tha Maasoremant of Power lamoi s oa
lahrall'a Watar-aapplj Raginaaring. 8to» 4 oe
Miall't Watar-powar. 8to, s 00
fnartaa'a Water and PabliaHaallh lamo, i so
Watar-flltratioa Woffca xamo, a so
OangoiUat and Ksttarii Oanaiml f ormohi for tha Uniform Flow U Water in
Rifari and Other Channalt (Bering and TTantwIna.) 8to, 4 00
Haian'to FUtnrtion of Pablia Watar^aapply 8to, 3 00
Haslehorift Towen and Tanka for Water-worka 9vo, a so
Hefaeheft 1x5 Ezperimanti on the Carrying Capacity of Large, Rlreted, Metal
Condvita 8to, a 00
Kaaon'a Water-aupply. (Conaideiad Principally from a Sanitary Stand-
point) 3d Bdition, Rewritten 8to, 4 00
Merriman'a Treatiae on Hydraulica. 9th Edition. Rewritten 8to, 5 00
• Michie'a Blementa of Analytical Machanica 8to, 4 00
Schuyler's Resenroira for Irrigation, Water-power, and Domeatic Water-
supply Large 8fo, 5 00
•^ Thomas and Watt* s Improramant of Riyers. (Poat, 44 c additional), 4to, 6 00
Tameaore and RuaselTs PnbHc Water-aupplias. 8to, 5 00
Wegmann'a Desicn and Construction of Dama. 4to, 9 00
Water-sopplyof tiha City of Raw Tork from 1658 to'x89S 4to,io 00
Wairtach's HydrauBca and Hydraulic Motors. (Du Bois.) 8to, s 00
WQson's Manual of Irrigation Rnginaering Small 8to« 4 oo
WoUTs Windmill aa a Prime MoTer 8to, 3 oa
Wood's TurUnes Sto, a go
Elements of Analytical Machanica Sro, 3 00
MATERIALS 09 BHOnfBBRIHO.
Baker's Treatiae on Maaonry Construction 8to, 5 00
Roads and PaTsments. 8to, 5 00
Bkck's United Stataa Public Wofka Obtong 4to, 5 00
Bofay's Strength of Materials and Theory of Structures. Sro, 7 50
Burr's Elastidtr and Resistance of tha Materials of Engineering. 6th Edi-
tion, Rewritten. .Svo, 7 90
Byrne's Highway Construction 8vo, s 00
Inspection of the Matarials and Workmanship Employed.in Construction.
z6mo, 3 00
Church's Mechaaica of Engineering 8to, 6 00
Du Boia's Mechanics of Engineering. VoL I Small 4to, 7 50
Johnson's Materials of Construction Large 8to, 6 00
Kaap^a Caat Iron Sro, a 50
Lansa's Applied Mechanica 8to, 7 so
Martena's Handbook on Testing Materials. (Henning.) a vols. 8to, 7 90
MarrUrs Stones for Building and Decoration 8to, 5 00
7
Digitized by VjOOQ IC
Mnriman*! T«zt-book on tht M^ch■nk« of Materialf 990, 4 oo
Strength of Materials lamo, x 00
MetcalTi Steel. A Manual for Stee^ttaers xamo* a 00
Patton't Practical Treatise on Fonndations 8to, 5 00
Richey's Hanbbook for Building Superintendents of Construction. (/» press.)
Rockwell's Roads and Parements in France zamo, i as
Sabin's Industrial and Artistic Ttitmologj of Paints and Varnish 8to, 3 00
Smith's Materials of Machines lamo* i 00
Snow's Prindiial Species of Wood 8to, 3 50
Spalding's Hydraulic Cement lamo, a 00
Text-book on Roads and Pavements i2mo» a 00
Taytor and Thompson's Treatise on Concrete, Plain and Reinforced. (In
Thurston's Materials of Engineering. 3 Parts Sto, 8 00
Part L— Non-metallic -Materials of Engineering and Metalhirgy Sro, a oo
Part n.— Iron and Steel Sto, 3 S*
Part m.— A Treatise on Brasses* Bronzes, and Other Alloys and their
Constituents Sro, a s*
Thurston's Text-book of the Materials of ConstructloA 8to» 5 ••
TlUson's Street PaTements and Paving Materials Svo» 4 00
WaddelTs De Pontibus. (A Pocket-book for Bridge Engineers.) . . x6mo, mot:* 3 00
Spedikations for Steel Bridges zamo, z ag
Wood's Treatise on the Resistance of Materials, and an Appendix on the Pree-
erration of Thnber Sro. a 00
Elements of Analytical Mechanics 8to» 3 00
Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . ,9f9, 4 o*
RAILWAY EKGIlfEERIirG.
Andrews's Handbook for Street Railway Engineexi. 3X5 inches, morocco, z ag
Berg's Buildings and Structures of American Railroads 4to» S 00
Brooks's Handbook of Street Railroad Location z6mo. morocco, z go
Butts's Civil Engineer's Fteld-book z6mo, morocco, a so
CrandaU's Transition Curve z6mo, morocco, z so
Railway and Other Earthwork Tables Svo, z so
Dawson's "Engineering" and Electric Traction Pocket-l^ok. i6mo, morocco, s oe
Dredge's History of the Pennsylvania Railroad: (1879) Paper* S 00
* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., as oe
Fisher's Table of Cubic Yards Cardboard. as
Godwin's Railroad Engineers* Field-book and Explorers' Guide. .... z6mo, mor., a so
Howard's Transition Curve Field-book i6mo, morocco, z so
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
bankments Svo, z 00
Molitor and Beard's Manual for Resident Engineers z6mo, z 00
Eagle's Field Manual for Railroad Engineers z6mo, morocco, 3 oo
Philbrick's Field Manual for Engineers z6mo, morocco, 3 00
Searles's Field Engineering i6mo, morocco, 3 00
Rsilroad SpiraL , . .z6mo, morocco, z 90
Tayk>r's Prismoidal Formula and Earthwork Svo, z so
• Trautwine's Method of CalcuUting the Cubic Contents of Excavations and
Embankments by the Aid of Diagrams. 8vo, a 00
The Field Practice of (Laying Out Circukr Curves for Railroads.
zamo, morocco, a so
Cross-section Sheet Paper, as
Webb's Railroad Construction, ad Edition, Rewritten z6mo. morocco, s oo
Wellington's Economic Theory of the Location of Railways Small Svo, s 00
DRAWING.
Barr's Kinematics of Machinery Svo, a so
o Bartletf s Mechanical Drawing Svo, 3 oo
o *• AbridgedEd. Svo, z «•
8
Digitized byVjOOQlC
CooUdflt't Manual of Diawlnc 8to, paper, i oo
CooUdga and FrMoum't Btomanti of Oanaral Dfafting f of ^r'^'*"'*-* Engl-
naen. (In prwt.)
Durley't Kinematica ol Machines. 8vo, 4 o»
Hill't Teit-book on Shadea and Shadows, and PerspectlTe 8vo, a 00
lamiion's Blements of Mechanical Drawlnc. (/n prsss.)
Jones's Mechine Desicn:
Part L — Kinematics of Machinery 8vo, i 90
Part n. — Form, Strength, and Proportions of Parts 8to» 5 00
MacCoffd's Elements of Descriptire Oeometr> Sro. 300
Kinematifls; or. Practical Mechanism. 8to, 500
Mechanical Drawing 4to, 400
Velocity Diagrams 8to, i 50
* Mahan's Descriptire Oeometry and Stone-cutting > Sro, i go
Industrial Drawing. (Thompson.) 9vo, 5 50
Moyer's DescriptiTe Oeometry. (In pre—.)
Reed's Topographical Drawing and Sketching 4to» 5 oo
Raid's Course la Mechanical Drawing Sto, 2 00
Text-book of Mechanical Drawing and Elementary Machine Design. .8to. 3 00
Robinson's Principles of Mechanism. 8to, 3 oo
Smith's Manual of Topographical Drawing. (McMillan.) 8fo, a 90
Warren's Elements of Plane and Solid Free-liand Geometrical Drawing.. lamo, x ••
Drafting Instruments and Operations lamo, x as
Manual of Elementary Projection Drawing. xamo, x 50
Manual of Elementary Problems in the Linear Perspective of Form and :«
Shadow. ^ xamo, i oo
Pfame Problema In Elementary Oeometry. xamo, i ag
Primary Oeometry xamo, 78
Elements of DescriptiTe Oeometry, Shadows, and PetipectiTe 8vo, 3 50
(General Problema of Shades and Shadows 8to, 3 oe
Elements of Machine Construction and Drawing Sro, 7 50
Problems. Theorems, and Examples in DescriptiTe Oeometry Sto, a 30
Weisbach's Kinemstics and the Power of Transmission. (Hermann and
Klein.) Sto, 5 00
Whelpley's Practical Instruction in the Art of Letter EngraTing xamo, a 00
Wilson's Topographic Surreying Sto, 3 SO
Free-hand PerspectlTe Sto, a so
Free-hand Lettwring. Sto, x 00
WoolTs Elementary Course in DescriptiTe Geometry Large Bto, 3 00
ELECTRICITY AND PHYSICS.
Anthony and Bracketf s Text-book of Physics. (Magie.) Small Sto* 3 00
Anthony's Lecture-notes on the Theory of Electrical Measurements xamo, x 00
Benjamin's History of Electricity Sto, 3 00
Voltaic CelL Sto, 3 00
Ckssen's QuantitatlTe Chemical Analysis by Electrolysis. (Boltwood.). .Sto, 3 00
Crehore and Squier's Polarizing Photo-chronograph 8to« 3 oe
Dawson's "Engineering" and Electric Traction Pocket-book.. i6mo, morocco, 5 00
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von
Ende.) xamo, a so
Duhem's Thermodynamics and Chemistry. (Burgess.) Sto, 4 00
Flather's DTuamometers, and the Measurement of Power xamo, 3 oo
Gilbert's De Magnate. (Mottelay.) Sto, a 50
Hanchetf 8 Attemating Currents Erplained. lamo, x 00
Hering's Ready Reference Tables (0>nTer8ion Factors) x6mo, morocco, a so
Hohnan's Precision of Measurements Sto, a 00
Telescopic Mirror-scale Method, Adjuatmenta, and Tests.. . . .Large Sto, 7S
9
Digitized by CjOOQ IC
Laadaoar't Spectmm AiiAlytii. (Tinfto.) 8to« s cm
LtChateBer'tHlch-tMBptnitiirtllMtartaMiitB. (l!o«i<mml Ituiii )iamo, s oe
Lgb'tBltctrotybMidBlecUinyiitiMdtofOfimicCoaipomidfc (LqreM.>umo, i oe
• LyoiWtTnfttiae on Btoctromagnttic Phtnomtna. Volt. L and IL )»?o, ondi, 6 oo
•mchk. Blemonti of Wnvo Motion RokUlnc to Seuttd and Light 8to, 4 ••
HiaodoftBlemontaryTnatiMonBlectficBatttclia. (FlahoadLi lamo, s 9*
• ltonnbtig*t Bloctrknl Kngtn— ring, (HiMano Oto— Kinthnamor.). . . .>yo, s 90
S9U,S6rris,andHoslt*tBlMtrlcnl]Inoliin«7* VbLL Sio, » flo
Thonton't Stationafy Staam-onglnea 8to, a 90
• TOlman'i Blamentary Laaiont In Heat 8to, i 9»
Toiy and Pttcher*! Manoal of Laboratory Phyalct Small 9fo, a oe
Ulka'a Modom Bloctrolytk Coppar Baflning .Sro, 3 oe
LAW.
• OaTia't Blemantt of Law Bro, a 90
• Trtatiia on tba Military Law of United Statea 8fo, 7 oe
• Sbeep, 7 Se
Manual for Conrta-martlal i6mo, morocco, i 9e
Walfa Sngineering and Architectural Joriapnidence 9fo, 6 oe
Sheep, 6 90
Law of Operationa Preliminary to Conatmetlon In Engineering and AicU-
tecture ■«•..> .«...•* ..• 8fo« see
Sheep, s 9e
LawofContraeta tvo, s oe
Wlnthrop'a Abridgment of MUHary Law , lame, a 9*
* MAHUFACTURBS.
Biffnadoo'a Smokeleaf Powder^Vltro-cellnloee and Theory of the CeUoloae
Molecule lamo. 2 90
BoOand'a Iron Founder lamo, a *9e
"The Iron Founder,** Supplement lamo, a 90
Bncydopedia of Founding and Dictionary of Foundry Terma Uaed in the
Practice of Moulding lamo, 300
' Biailer*a Modem High Bzploaivea Sro, 4 oe
'BflronfaBnaymea and their Applicatlona. (Pnacott) Sro, 300
Flt«gerald'a Beaton Machinlat xSmo, i oe
Fofd'to Boiler Making for Boiler Maken iSmo, x oe
BopUna'a.Oil-cbemiata' Handbook Sro, 3 ee
Keep** Caat Iron. Sro, a pe
Leaeh'a The Inapection and Analjala of Food with Special Reference to State
ControL (In pfporalien.)
MatdiewB*a The TeitUe Ftbrea. (In pnee.)
MetcaVft SteeL A Manual for SteeKuaera xame, a ee
MetcaBe^a Coat of Manufacturea^And the Admlnlatrmtlon of Workahopa,
Public and Private Svo, 5 oe
Meyer'a Modem LocomotiTe Conatruction 4to, le ee
Moiae'a Cakulationa uaed in Cane eugar Fiactoriea. xtew, OMMOcoe, 1 9*
• Beiaig'a Guide to Piece-dyeing 9wo, as ee
Sabin*a Indnatrlal and Artiatic Techn^ogy of Palnti and Yamiah S?o» p 00
Smith'a Pruai working of Metale .Sve, 3 ee
Spaldlng*a Hydraulic Cement. xamo, a oe
Spencer'a Handbook for Chemiata of Beet-eugar Bouaee ittmo, morocco, 3 oe
Handbook tor ougar Manutacturera ana their Chemiata.. . i6mo, morocco, a ee
Taylor and Thompion'a Tnatiae on Concrete, Plain and Balnfteced. (In
ThefBlen'to Manual of Steam-boUen, their Deaigna, Coaatructien and Opeim-
tloQ Svo, see
10
Digitized byVjOOQlC
* Waftt^t L«ct«ffM OB Biplodrtt 8to» 4 «•
W«ft Anwriwn f ooodiy FnctlM lamo, a 5»
]foolter*t Ttst-book lamo. a 9*
Wlaehmanii't Sugar Analjiia. Small 8to, a 90
Worn Windmill aa a Prima lloTtr Sro, 3 00
Woodbnry'a FIra Proteetion of MUb Svo, a &•
Wood*a RnatltM Coatings: Corroaion and BI>ctro|yala of lion and Stael. . .8fO, 40a
MATHEMATICS.
Bakar's BUiptic Fnnctiona Sro, i 9*
* Baat's Elemanti of Diilarantial Calcqlaa lamo, 4 oo
Briggi^ Blamania of Plana Analftle Oaomatry xamo, t 00
CompCon*! Manoal of Logarithmic Compotationa xamo, i 9a
OaTit'a Inttodoctlon to tha Logk of Algabra Sra, x so
• Dickson'i Colkga Algabra Largo lamo, x 9*
* Answers to Dickson's Colkga Algabra Sro, paper, 2%
• Introduction to the Theory of Algebraic Bqoationa Large lamo, x ag
HalMed*a Blamants of Geometry 8to» x 75
Elementary Syntlietic Geometry 8to, x 90
Rational Geometry. xamo,
• Johnaon's Three-place Logarithmic Tablea: Veat-pocket siie paper, xg
xoo copies for 9 00
• Mounted on heavy cardboard, 8X 10 inchea, ag
xo copiea for a 00
Elementary Tnatiae on the Integral Calcoloa Small 8?o, x ga
Cunre Tracing in Cartesian Co-ordinatea xamo, x 00
Treatise on Ordinary and Partial Differential Bqvationa Small 8?o, 3 ga
Tlieory of Brrors and the Method of Least S^oarea xamo, x 50
* Theoretical Mechanics xamo, 300
BIplace's PhUoaophlcal Basay on Probabifitiea. (Tmsoott and Emory.) xamo, a oa
* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other
Tablea 8to, 3 00
Trigonometry and Tablea pabUahed separately Each, a 00
* Lodlow'a Logarithmic and Trigonometric Tables 8to, x oa
Maurer's Technical Mechanics. 8to, 4 oe
Msiriman and Woodward's. Higher Mathematica 8to, 9 oa
Meiriman's Method of Least Squares. 8to, a 00
Rice and Johnson's Elementary Treatiae on the Differential Calculus. 8m.,8T0, 3 00
Differential and Integral Cakuhis. a fols. In one Small 8to, a 90
Sabin'a Induatrial and Artistic Technology of Paints and Varnish. 8?o, 3 «•
Wood's Elements of Co-ordinate Geometry 8vo. a 00
Trigonometry: Analytical, Plane, and Spherical xamo, x 00
MBCHAHICAL BHODIBERDIO.
MATERIALS OF BROIBBERIirO. STBAM-EHGHTES A5D BOILERS.
Bacon's Forge Practice xamo, x ga
Baldwin's Steam Heating for Buildings xamo, a 90
Barr's Kinematics of Machinery 8to, a 90
• Bartletf s Mechanical Drawing Svo, 3 00
• *• -.44 AbridgedEd. Syo. x 90
Benjamin's Wrinkles and Recipes xamo, a oa
Carpenter's Experimental Engineering Svo, 6 00
'^ Heating and Ventilating Buildinga Sto, 4 00
Caryt Smoke Suppression In Plants using Bituminous CoaL (/n pt§p-
&tation,y
dsck's Gas and Oil Engine Small Sto, 4 00
Oaoldge's Manual of Drawing Sro, paper, xoo
11
Digitized by VjOOQ IC
CooUdge and Freemmn's Elementi of General Dxaftinc for Ifocbanical Bft-
gineert. {hn pre$$.)
Cromwell't Treatise on Toothed Gearing lamo. i 5»
Treatise on Belts and PuUeys. xamo» s fl*
Darky's EJnematics of Machines 8to, 4 jm
TIather's Drnamometers and the Measurement of Power zamo, 3 00
Rope DriTing lamo, a 00
OUTS Gas and Fuel Analysis for Engineers e lamo, x as
Hall's Car Lishrication. lamo, x 00
Htring's Ready Reference Tables (Conversion Factors) x6mo, morocco* a 50
Button's The Gas Engine 8to, 5 00
Jones's Machine Design:
Part I. — Kinematics of Machinery 8to, t flo
Part n. — Form, Strength, and Proportions of Parts. 8f<ft, 3 00
Kent's Mechanical Engineer's Pocket-book s6mo» morocco, 5 00
Kerr's Power and Power Transmission 8?o« a 00
Leonard's Machine Shops. Tools, and Methods, (In pr999,)
MacCord's Kinematics; or. Practical Mechanism. 8?Ot 5 00
Mechanical Drawing 4lOt 4 00
Velocity Diagrams 8to, x 50
Mahan's Industrial Drawing. (Thompson.) 8to. 3 50
Poole's Calorific Power of Fuels , Sro. 3 00
Reid's Course in Mechanical Drawing 8to. a 00
Text-book of Mechanical Drawing and Elementary Machine Design. .8to, 3 00
Richards's Compressed Air xamo, x so
Robinson's Principles of Mechanism. '. . .8to, 3 00
Schwamb and Merrill's Elements of Mechanism, (/n preat.)
Smith's Press-worlcing of Metals 8vo, 3 o»
Thurston's Treatise on Friction and Lost Work in Machinery and Mlil
Work. 8to, 3 «0
Animal as a BCachine and Prime Motor, and the Laws of Energetics . xamo, x fb
Warren's Elements of Machine Construction and Drawing 8fo, 7 50
Weisbach's Kinematics and the Power of Trax^smission. Herrmann —
Klein.) 8to, s 00
Machinery of Transmission and Governors. (Herrmann — Klein.).. 8to, 5 00
Hydraul.cs and QydrauUc Motors. (Du Bois.) 8to, 5 00
Wolff's Windmill as a Prime Mover. 8to, 3 00
Wood's Turbines Svo, a 50
MATERIALS OF EH OINBERING.
Bovey's Strength of Materiab and Theory of Structures 8vo, 7 99
Burr's Elasticity and Reaist^nce of the Materiab of Engineering. 6th Edition,
Reset 8to, 7 80
Church's Mechanics of Engineering Svot 6 oo
Johnson'* Materials of (instruction Larga 8vo, 6 00
Keep's Cast Iron 8vo» a so
Lanza's Applied Mechanics 8vo, 7 so
Martens's Handbook on Testing Materials. (Henning.) 8to, 7.50
Merriman's Text-book on the Mechanics of Materiab 8to, 4 00
Strength of Materab xamo, x 00
MetcaVs SteeL A Manual for Steel-users xamo. a 00
Sabin's Industrial and Artistic Technology of Paints and Varnish. 8to, 3 00
Smith's Materiab of Machines xamo, x 00
Thurston's Materiab of Engineering 3 Tob • Sto, 8 00
Part IL— Iron and Steel 8to, 35a
Part in. — A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents. .8yo » 50
Text-book of the Materiab of Construction. 8vo« g 00
12
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Wood's TreatiM on tho Redstenco of MAteriali and an Appendix on the
Prtterratlon of Timber 8vo, a oo
Blsments of Analytical Mechanic! 8to, 3 00
Wood'i Rottleaa Coatingi: Corrotion and Blectrolyiit of Iron and SteeL . .8to, 4 00
STBAM-SNOniBS AHB B0IL!BRS.
Camof 8 Reflectiona on the Motire Power of Heat. (Thurston.) lamo. x so
Dawion't '^Engineering^ and Electric Traction Pocket-book. .i6nio, mor.» s 00
Ford'i Boiler Maldng for Boiler Makers i8mo, i 00
0008*8 Locomottye Sparia 8vo, a 00
Bemenway's Indicator Practice and Steam-engine Economy lamo, a 00
Button's Mechanical Engineering of Power Plants 8vo, 5 00
Beat and Beat-enginea 8to, 5 oo
Kenfs Steam-boUer Economy 8to, 4 00
Kneaas's Practice and Theory of the Injector 8to x go
MacCord's SHde-Talrei 8to, a 00
Meyer's Modem LoeomotiTe Construction 4to, 10 00
Peabody's Manual of the Steam-engine Indicator xamo, x so
Tables of the Propertiee of Saturated Steam and Other Vapors Svo. x 00
Thermodynamics of the Steam-engine and Other Beat-engines 8to» 5 00 ^
Vahre-gears for Steam-enginee 8to, a 50
Peabody and Miller's Steam-boilers 8?o, 4 oo
Prey's Twenty Years with the Indicator Large 8?o, a so
Pupln's Thermodynamics of Reyersible Cycles In Oases and Saturated Vapors.
(Osterberg.) xamo, x ag
Reagan's Locomotives : Simple, Compound, and Electric xamo, a 50 1
Rontgen's Principles of Thermodynamics. (Du Bois.) 8to, $00
Sinclair's LoeomotiTe Engine Running and Management xamo, a 00
Smart's Handbook of Engineering Laboratory Practice xamo, a 50
Snow's Steam-boiler Practice ^ 8?o, 3 oo
Spangler's ValTe-gears 4 .". . . .Sro, a &•
Notes on Thermodynamics xamo, x 00 4
Spangler, Oreene, and MarshalTs Elements of Steam-ongineering 8?o, 300
Thurston's Bandy Tables 8to. x 50 i
Manual of the Steam-engine a vols. Sto, xo 00
Part L — Bistory. Structuce, and Theory 8to, 6 00
Part n. — Design, Construction, and Operation Sro, 6 00
Bandbook of Engine and Boiler Trials, and the Use of the Indicator and
the Prony Brake Sto 5 00
Stationary Steam-engines Sto, a go
Steam-boiler Explosions in Theory and in Practice xamo x 50
Manual of Steam-boilerf, Their Designs, Construction, and Operation. Sto, s 00
Weisbach's Beat, Steam, and Steam-engines. (Du Bois.) Sto, 500
Whitham's Steam-engine Djsign Sto, 5 00
Wilson's Treatise on Steam-boilers. (Flather.) x6mo, a go
Wood's Thermodynamics Beat Motors, and Refrigerating Machines. . . .Sto. 4 00
MBCHABICS AHD MACHIIIERT.
Barr's K*pif'"«»^''^ of Machinery Sto, a 9a
BoTey's Strength of Materials and Theory of Structures Sto, 7 90
Ohase's The Art of Pattern-making xamo, a so
Chordal. — Extracts from Letters xamo. a 00
Church's Mechanics of Engineering Sto, 6 00
Rotes and Examplee in Mechanics Sto, a 00
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Com^n't First L«fioiit in Matal-woridiic , lamo,
Compton and Dt 6roodt*t Hm Spetd Latht ...,.'. ,^ . iamo»
! CromweU't Trestitt on TootM Otarinc ^\ i9IDO,
Treatise on Belts and PoOeys .z3^t
Dana's Text-book of Slementary Mechanics for the Use of CoHeges and
{ Scbooli , zamo,
' Dinger's Machinery Pattern MaUa^. . ,. «*..... ^. tamo.
Dredge's Record of the Transportation Bzhlbits Boilding of the World's
Miupbiao g^posHionof 18^...^^*, 4to, half nia;roc€o»
Dti Bolt's Bkpsntary Principles of Mechanics;
Vol* tr-ll^Mi»nuitU» ^ ....*.. 9^
XoL n.— Statics. , ^,....8»o^
yoLin«~Kin^c«....,., , ...^.Sro*
Mechanics of Bngineerinc. VoL I...... Small ^to,. ~
^VoLIL... .^.. Small 4to,
Dvrley's Kinematics of Mschtnes' , .870,
flOgerald's Boston Machinist \,^. z6mo,
Flatber's Dynamometers, anA the Measurement of Power. zamo«
Rope DtiTing xamo,
Ooes's LoeomotiTe Sparks. , .* p.. Sto
Hall's Car Z«obricatki9« , «* % « .iamo»
MpUy's Act of Saw Filii«„ ,.....«... zSmo.
< Jolytloa^s Theoretical Mechanics. zamo.
Statics hy Graphic and Algebraic Mtthoda .8to.
Jess's Machine Desifn:
1 ' Part L— f inematirs of Machinery « 8to,
[ PartIL— Form, Stwngth*rand Proportions of Parts, ........ .^« .. . .Sins^
' Kin's Power aftd Power Trsnsmlsikm , . ^ Sro.
I,ttnta*s Appfied Mechantes. . . . .u A ..«.. « Sro,^
* Leonard s Machine Shops. Tods, tnd Jfetbads. (/• prm$.)
MacCord's Kinematics; or, Practtsal Mechanism Svo,
Velocity Dia^fama 8fo,
; Minrer's Technical Mechanics. Sto,
't Mazyiihan's Text-book on the Mechanics of MttiriAli. *...... Svo,
'•MkhysKleznents of AnalytkalMsghf tra... ....... ^>,,^>^, .fvo,,
>K iUasan's LocomotiTss: Simple. Compoond, and Electric • ^tMO^
^ Rei^s Course in Mechanical Drawing Sro.
Text-book of Meshanirat Drawiikg and Btomonlary MaclMlM Design . . Sro.
RIchardb's. Compressed Air.... .^att^
. RoWneoiqo Principles of Mechmlsm ,,,, ,,,.... ^.'.Srq,.
Ryan. Sorris, and Hoxie'sElsctrkalMacUnery. VoLI.... ....8«o.
* SchiNimt) and Merrill's Blements of Mechanism. (Mprsw.)
1 Sin^titA Locomotire-engine Rtmning and Management. . . .% .^ . . > . .\ i jkiaifc
I Smithl Pfvss-wortting of Metals « .tn>»
T MAferials of Machines ... .....«..«. bm^
^^Sj^^lSf. Greene, and MarstialTs Blsments of 8leam-eoginssrim.«.A ..,8w#
ttav^atoo's treatise on Friction and Lost Work in Machinsfy mad WOi
J^ Work *^^ ilOr
% AnimalasalMchine aiid Prime Motor, and the Laws of Energetics. lamo.
i Wasrll»> tlemen|B of- Machine Construction and Drawing Sro. >
\ W2!k«ch'4 Kiiiomatlo and tho Power of Tranwnission, jiVfltmma^
\ - Klein.) ;....... Sro.
\ l^chinery of Transmission and GoT^z)prs. (Berrmann — KIeln.f.8T0.
] Wocsl's filaments of Analytical Mechanics'. , Sro,
I Pi^ciplis of Elementary Mechanics..... ^ lamo.
Turbines ....*..... .Svo.
^ The World's Cohimbian Exposition of z8o3....^r 4to.
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